Harmonic Analysis, Geometric Measure Theory and Additive Combinatorics

Home , Geometric measure theory

Summer School,∗ Catalina Island Jun 24th - Jun 29th 2012

Organizers:

Izabella Laba, University of British Columbia, Vancouver

Malabika Pramanik, University of British Columbia, Vancouver

Christoph Thiele, University of California, Los Angeles

∗supported by NSF grant DMS 1001535

1 Contents

1 Arithmetic progressions in sets of fractional dimension 6 Gagik Amirkhanyan, Georgia Tech ...... 6 1.1 Introduction ...... 6

2 Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, Part II: Applications 11 Michael Bateman, UCLA ...... 11 2.1 Introduction ...... 11 2.2 Establishing the uncertainty principle from Turan’s lemma: an idealized scenario ...... 13 2.2.1 Proof of Theorem 6 given wishlist above ...... 14 2.3 Averaging ...... 15

3 The endpoint case of a Stein-Tomas theorem for subsets of the real line 17 Marc Carnovale, UBC ...... 17 3.1 Introduction ...... 17 3.2 Lions-Peetre Interpolation Spaces ...... 19 3.3 Skirting the Triangle Inequality via Interpolation ...... 21 3.4 Finishing up ...... 22

4 Necessary conditions for Lp(Rn)-Fourier multipliers 26 Vincent Chan, UBC ...... 26 4.1 Introduction ...... 26 4.2 Idempotents case ...... 26 4.3 A generalization ...... 28

5 Salem-Bluhm’s construction of Salem sets 31 Xianghong Chen, UW-Madison ...... 31 5.1 Introduction ...... 31 5.2 The main result ...... 31 5.3 The set ...... 32 5.4 The measure ...... 33 5.5 The Fourier transform ...... 33 5.6 Randomization ...... 33

2 5.7 From average decay to deterministic decay ...... 33 5.8 The key estimate ...... 34 5.9 Proof of the average decay ...... 35 5.10 The dimensions ...... 36

6 Buﬀon’s needle estimates for rational product Cantor sets 37 Kyle Hambrook, UBC ...... 37 6.1 Introduction ...... 37 0 6.2 The SSV property of φt ...... 40 00 6.3 The SLV structure of φt ...... 41 6.4 Reduction to lower bounds on integrals; required upper bounds 43 6.5 The main argument ...... 44

7 Projecting the One-Dimensional Sierpinski Gasket 47 Edward Kroc, UBC ...... 47 7.1 Introduction ...... 47 7.2 The measure of the projections ...... 48 7.3 Bounds on the dimension of the projections ...... 50

8 Wiener’s ‘closure of translates’ problem and Piatetski-Shapiro’s uniqueness phenomenon 53 Allison Lewko, University of Texas at Austin, Microsoft Research New England ...... 53 8.1 Introduction ...... 53 8.2 Tools ...... 55

9 Bounded orthogonality systems and the Λ(p)-set problem II 57 Mark Lewko ...... 57 9.1 Introduction ...... 57

10 Local estimates of exponential polynomials and their applications to inequalities of uncertainty principle type - part I 62 Christoph Marx, UCI ...... 62 10.1 Introduction ...... 62 10.2 Nazarov’s theorem ...... 64 10.2.1 Bernstein-type estimates and order reduction . . . . . 64 10.2.2 The role of “zero counting” ...... 65

3 10.3 Extensions ...... 66

11 Salem sets and restriction properties of Fourier transforms 68 Eyvindur Ari Palsson, University of Rochester ...... 68 11.1 Classical restriction ...... 68 11.2 Two notions of dimension ...... 68 11.3 Main result ...... 69 11.4 Salem sets ...... 70 11.5 Sketch of proof of theorem ...... 71

12 Maximal operators and diﬀerentiation theorems for sparse sets: Part II 73 Alex Rice, UGA ...... 73 12.1 Introduction ...... 73 12.2 Main results ...... 74 12.3 Motivation ...... 75 12.4 Linearization and discretization ...... 75 12.5 Transverse correlations ...... 76

13 Maximal operators and diﬀerentiation theorems for sparse sets, part I 79 Pablo Shmerkin, Surrey ...... 79 13.1 Introduction ...... 79 13.2 Construction of the sets Sk ...... 80 13.3 Internal tangencies and transversal intersections ...... 81 13.4 Main result ...... 82 13.5 A martingale argument, and conclusion of the proof ...... 83

14 Bounded orthogonality systems and the Λ(p)-set problem I 85 Stefan Steinerberger, University Bonn ...... 85 14.1 Introduction ...... 85 14.2 Sketch of the general proof ...... 87

15 On a problem of Erd˝oson sequences and measurable sets, & Inﬁnite patterns that can be avoided by measure. 89 Krystal Taylor, UMN ...... 89 15.1 Introduction: known classes of non-universal sets ...... 89 15.2 Results ...... 90

4 15.2.1 Sequences with ’slow decay’ are universal ...... 90 15.2.2 Inﬁnite sets are ’almost everywhere’ universal . . . . . 91 15.3 Sketch of proofs ...... 91 15.3.1 Construction of the set E in Theorem 2 ...... 91 15.3.2 Construction of the set E in Theorem 3 ...... 92

16 Averages in the plane over convex curves and maximal operators 95 Joshua Zahl, UCLA ...... 95 16.1 Introduction ...... 95 16.1.1 Background ...... 95 16.1.2 New Results ...... 96 16.2 Proof the Theorem 1 ...... 97 16.2.1 Reduction to a geometric problem ...... 97

5 1 Arithmetic progressions in sets of fractional dimension

after IzabellaLaba and Malabika Pramanik [1] A summary written by Gagik Amirkhanyan

Abstract

Let E ⊂ R be a closed set of Hausdorﬀ dimension α. We prove that if α is suﬃciently close to 1, and if E supports a probabilistic measure obeying appropriate dimensionality and Fourier decay conditions, then E contains non-trivial 3-term arithmetic progressions.

1.1 Introduction From the introduction of the summarized paper: “

Deﬁnition 1. Let A ⊂ R be a set. We will say that A is universal for a class E of subsets of R if any set E ∈ E contains an aﬃne (i.e. translated and rescaled) copy of A.

If E is the class of all subsets of R of positive Lebesgue measure, then it follows from Lebesgue’s theorem on density points that every finite set A is universal for E. Namely, let E have positive Lebesgue measure, then E has density 1 at almost every x ∈ E. In particular, given any δ > 0, we may choose an interval I = (x − , x + ) such that |E ∩ I| ≥ (1 − δ)|I|. If δ was chosen small enough depending on A, the set E ∩ I will contain an affine copy of A. An old question due to Erd˝osis whether any infinite set A ⊂ R can be universal for all sets of positive Lebesgue measure. It is known that not all ∞ infinite sets are universal: for instance, if A = {an}n=1 is a slowly decaying sequence such that a → 0 and an−1 → 1, then one can construct explicit n an Cantor-type sets of positive Lebesgue measure which do not contain an affine copy of A [2]. There are no known examples of infinite sets A which are universal for the class of sets of positive measure. In particular, the question −n ∞ remains open for A = {2 }n=1. The purpose of this paper is to address a related question: if A ⊂ R is a finite set and E ⊂ [0, 1] is a set of Hausdorff dimension α ∈ [0, 1], must E contain an affine copy of A? In other words, are finite sets universal for

6 the class of all sets of Hausdorff dimension α? This more general statement already fails if A = {0, 1, 2} and E is a set of Hausdorff dimension 1 but Lebesgue measure 0. This is due to Keleti [3], who actually proved a stronger result: there is a closed set E ⊂ [0, 1] of Hausdorff dimension 1 such that E does not contain any “rectangle” {x, x + r, y, y + r} with x 6= y and r 6= 0. Hence one may ask if there is a natural subclass of sets of fractional dimension for which a finite set such as {0, 1, 2} might be universal. This question is addressed in Theorem 2, which is the main result of this article. We define the Fourier coefficients of a measure µ supported on [0, 1] as Z 1 −2πikx µb(k) = e dµ(x). 0 Theorem 2. Assume that E ⊂ [0, 1] is a closed set which supports a probabilistic measure µ with the following properties: α (A) µ([x, x + ]) ≤ C1 for all 0 < ≤ 1, −B − β (B) |µb(k)| ≤ C2(1 − α) |k| 2 for all k 6= 0, where 0 < α < 1 and 2/3 < β ≤ 1. If α > 1 − 0, where 0 > 0 is a sufficiently small constant depending only on C1,C2, B, β, then E contains a non-trivial 3-term arithmetic progression. We note that if (A) holds with α = 1, then µ is absolutely continuous with respect to the Lebesgue measure, hence E has positive Lebesgue measure. This case is already covered by the Lebesgue density argument. In practice, (B) will often be satisfied with β very close to α. It will be clear from the proof that the dependence on β can be dropped from the statement of the theorem if β is bounded from below away from 2/3, e.g. β > 4/5; in such cases, the 0 in Theorem 2 depends only on C1,C2,B. The assumptions of Theorem 2 are in part suggested by number-theoretic considerations, which we now describe briefly. A theorem of Roth states that if A ⊂ N has positive upper density, i.e. #(A ∩ {1,...,N}) lim > 0, (1) N→∞ N then A must contain a non-trivial 3-term arithmetic progression. Szemerédi’s theorem extends this to k-term progressions. It is well known that Roth’s theorem fails without the assumption (1). However, there are certain natural cases when (1) may fail but the conclusion of Roth’s theorem still holds.

7 For example, there are variants of Roth’s theorem for random sets and sets such as primes which resemble random sets closely enough. The key concept turns out to be linear uniformity. It is not hard to prove that if the Fourier coefficients Ab(k) of the characteristic function of A are sufficiently small, depending on the size of A, then A must contain 3-term arithmetic progressions even if its asymptotic density is 0. The Roth-type results mentioned above say that the same conclusion holds under the weaker assumption that A has an appropriate majorant whose Fourier coefficients are sufficiently small (this is true for example if A is a large subset of a random set). If the universality of A = {0, 1, 2} for sets of positive Lebesgue measure is viewed as a continuous analogue of Roth’s theorem, then its lower- dimensional analogue corresponds to Roth’s theorem for integer sets of density 0 in N. The above considerations suggest that such a result might hold under appropriately chosen Fourier-analytic conditions on E which could be interpreted in terms of E being “random.” We propose Assumptions (A)-(B) of Theorem 2 as such conditions. To explain why Assumptions (A)-(B) are natural in this context, we give a brief review of the pertinent background. Let dimH (E) denote the Hausdorff dimension of E. Frostman’s lemma (see e.g. [2]) asserts that if E ⊂ R is a compact set then

dimH (E) = sup{α ≥ 0 : ∃ a probabilistic measure µ supported on E

such that (A) holds for some C1 = C1(α)}. We also deﬁne the Fourier dimension of E ⊂ R as

dimF (E) = sup{β ≥ 0 : ∃ a probabilistic measure µ supported on E −β/2 such that |µb(ξ)| ≤ C(1 + |ξ|) for all ξ ∈ R}, R −2πiξx where µb(ξ) = e dµ(x). Thus (A) implies that E has Hausdorﬀ dimension at least α, and (B) says that E has Fourier dimension at least 2/3. It is known that

dimF (E) ≤ dimH (E) for all E ⊂ R; (2) in particular, a non-zero measure supported on E cannot obey (B) for any β > dimH (E). It is quite common for the inequality in (2) to be sharp: for instance, the middle-thirds Cantor set has Hausdorﬀ dimension log 2/ log 3,

8 but Fourier dimension 0. Nonetheless, there are large classes of sets such that dimF (E) = dimH (E). Such sets are usually called Salem sets. Assumptions (A)-(B) are closely related, but not quite equivalent, to the statement that E is a Salem set. On the one hand, we do not have to assume that the Hausdorff and Fourier dimensions of E are actually equal. It suffices if (B) holds for some β, not necessarily equal to α or arbitrarily close to it. On the other hand, we need to control the constants C1,C2,B, as the range of α for which our theorem holds depends on these constants. Thus we need to address the question of whether measures obeying these modified assumptions can actually exist. We prove that given any C1 > 1, C2 > 0 and 0 < β < α < 1, there are subsets of [0, 1] which obey (A)-(B) with B = 0 and with the given values of C1,C2, α, β. Our construction is based on probabilistic ideas similar to [4], but simpler. Salem’s construction [4] does not produce explicit constants, but we were able to modify his argument so as to show that, with large probability, the examples in [4] obey (A)-(B) with B = 1/2 and with C1,C2 independent of α for α close to 1. The key feature of our proof is the use of a restriction estimate. Restric- tion estimates originated in Euclidean harmonic analysis.” Our proof of Theorem 2 extends the approach of [5], [6] to the continuous setting of sets of fractional dimension for which a restriction estimate is available. We use of the trilinear form Λ in a Fourier representation and a decomposition of the measure µ into “random” and “periodic” parts.

References

[1] I. Laba, M. Pramanik Arithmetic progressions in sets of fractional dimension. Geom. Funct. Anal. 19 (2009), 429-456.

[2] K. Falconer, On a problem of Erd˝oson sequences and measurable sets, Proc. Amer. Math. Soc. 90 (1984), 77-78.

[3] T. Keleti, A 1-dimensional subset of the reals that intersects each of its translates in at most a single point, Real Anal. Exchange 24 (1998/99), no. 2, 843–844.

9 [4] R. Salem, On singular monotonic functions whose spectrum has a given Hausdorﬀ dimension, Ark. Mat. 1 (1950), 353–365.

[5] B. Green, Roth’s theorem in the primes, Ann. Math. 161 (2005), 1609- 1636.

[6] B. Green, T. Tao, Restriction theory of the Selberg sieve, with applications, J. Th´eor.Nombres Bordeaux 18 (2006), 147–182.

Gagik Amirkhanyan, Georgia Tech email: [email protected]

10 2 Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, Part II: Ap- plications

after F. Nazarov [1] A summary written by Michael Bateman

Abstract We use Turan’s lemma as proved by Nazarov to prove a quantitative version of the uncertainty principle, generalizing a result of Amrein and Berthier.

2.1 Introduction Nazarov’s paper [1] proves a variant of Turan’s lemma and then uses this lemma to establish several forms of the uncertainty principle. The uncertainty principle can be loosely stated as

Theorem 1 (Uncertainty principle, fuzzy version). A function cannot be simultaneously localized in space and in frequency.

An easy-to-establish version of this principle is

2 Theorem 2. For a smooth function f ∈ L (R) with ||f||2 = 1, we have Z Z 2 ˆ 2 |xf(x)| dx · |ξf(ξ)| dξ & 1. (1)

Already we can see from this inequality that if a function has spatial support in an neighborhood of the origin, then its Fourier support cannot 1 be contained in a C neighborhood of the origin. The inspiration for the version given by Nazarov is due to Amrein and Berthier:

Theorem 3 (Amrein-Berthier). Suppose f ∈ L2(R). If supp(f) and supp(fˆ) have ﬁnite measure, then f = 0 almost everywhere.

This summary focuses on Nazarov’s proof of the following quantitative version of this last theorem:

11 Theorem 4 (Nazarov). Let E, Σ ⊆ R have ﬁnite Lebesgue measure. Then Z Z 2 C|E||Σ| 2 ˆ 2 ||f||2 . e |f| + |f| . (2) R\E R\Σ This theorem can be applied to obtain precise estimates on the simulta- neous rates of decay for a function and its Fourier transform. For example:

1 1 Corollary 5. Fix p and q satisfying p + q = 1. Suppose f : R → R satisﬁes p q C1|x| ˆ C2|x| |f(x)| . e and |f(ξ)| . e . Then if C1 and C2 are small enough, f = 0 almost everywhere.

This is essentially a result of Morgan, who established optimal values for C1 and C2. We can obtain corollary by applying Nazarov’s theorem to sets ˆ of the form E = {x: |f(x)| ≥ λ1} and Σ = {ξ : |f(ξ)| ≥ λ2}. Before we begin with details of the proof, we present a simple reduction. It suﬃces to prove

Theorem 6. Let E ⊆ R have ﬁnite measure. Then for any function f ∈ L2(R) supported on E, we have Z Z ˆ 2 C|E||Σ| ˆ 2 |f| . e |f| (3) Σ R\Σ

Proof that Theorem 6 implies Theorem 4 . Let f ∈ L2(R) be arbitrary, and write fE := f1E. Then Z Z 2 2 2 ||f||2 = |fE| + |f| (4) R\E Z Z Z ˆ 2 ˆ 2 2 = |fE| + |fE| + |f| (5) Σ R\Σ R\E Z Z C|E||Σ| ˆ 2 ˆ 2 . e |f| + |fE| . (6) R\Σ R\Σ The second equality is by Plancherel and the inequality follows from Theorem 6. We now focus our attention on proving Theorem 6.

12 2.2 Establishing the uncertainty principle from Turan’s lemma: an idealized scenario In this section we establish Theorem 6 using Turan’s lemma, provided we have several favorable hypotheses at our disposal. Technically, these hypotheses are not justified; nevertheless we will be able to arrange for essentially equivalent hypotheses that are justified. To do this we will use the random periodization procedure described in Section 2.3. The goal is to understand the form of the argument rather than all of the details. To begin with, we fix a function f. We define the periodization g defined on the circle as follows: X g(t) = f(k + t). (7) k∈R An important fact is the following: ˆ Proposition 7. With g defined as above, we have f(m) =g ˆm. (Here we use the notation gˆm to emphasize the different domains of g and f.). Remark 8. This relationship is an important reason the function g is of interest to us. Another reason is the applicability of Turan’s lemma to exponential polynomials. The function f, by Fourier inversion, is an integral of exponential functions, but g is a sum of exponential functions. By considering an appropriately defined truncation of this sum, we will find a new function p, still related to f, that is a finite sum of exponential functions. Because of this form, we will be able to apply Turan’s lemma to p. We decompose the function g into two pieces: g = p + q, where

X 2πimt p(t) = gˆme . (8) m∈Σ∪{0}

In other words, p is the projection of g onto frequencies in Σ ∪ {0}. Of course this means that q is projection of g onto frequencies not in Σ ∪ {0}; this, together with the proposition above, suggests that q should have some R ˆ 2 relationship with the quantity |fE| on the right-hand side of Theorem R\Σ 6. We now write down a list of assumptions under which we can prove an estimate of the form in Theorem 6:

1 • WISH 1: |E| ≤ 10

13 • WISH 2: #(Z ∩ Σ) ∼ |Σ| 2 R ˆ 2 • WISH 3: ||q|| |fE| . 2 . R\Σ The ﬁrst assumption is rather mild; it will be handled essentially by a rescaled version of the argument below. The second and third assumptions are serious. The second says that the spectrum Σ should essentially be independent of the integers; the third is saying that the behavior of fˆ on the integers should capture the behavior of fˆ on R. As previously stated, this is quite a wishlist. We hint at how to realize these wishes in the ﬁnal section.

2.2.1 Proof of Theorem 6 given wishlist above We will actually prove the following stronger claim: Proposition 9. For all y ∈ R, we have Z ˆ 2 |Σ| ˆ 2 |f(y)| . C |fE| . (9) R\Σ Remark 10. Integrating this inequality over y ∈ Σ is enough to establish the result, since integrating over Σ costs a factor of |Σ|, which is easily absorbed by a quantity of the form eC|Σ|. In fact, we will establish the proof for y = 0, but applying the same proof to a modulation of f (and hence a translation in Fourier space) yields the claim for general y).

Pn 2πiλj t Lemma 11 (Simple case of Turan’s lemma). Suppose p(t) = j=1 e , λj ∈ Z. Suppose |p(t)| ≤ X for t in a set of measure ≥ α. Then X C |pˆ0| ≤ |pˆ0| . e α X. (10) k∈Z Now deﬁne sZ ˆ 2 F = {t: g(t) = 0} ∩ {t: |p(t)| . |fE| }. (11) R\Σ Notice on this set, p(t) = −q(t). Hence By Turan’s lemma, we have ˆ 2 2 2 |f| = |gˆ0| = |pˆ0| (12) !2 deg(p) X 2 ≤ |pˆ0| ≤ C |F | sup |p(t)| (13) t∈F k |Σ| Z |F | ˆ 2 ≤ C |fE| . (14) R\Σ

14 The second-to-last inequality follows from Turan’s Lemma, and the last inequality follows from WISH 2, which guarantees that the degree of the polynomial p is approximately |Σ|. (Speciﬁcally, deg(p) = #{Z ∩ Σ} ∼ |Σ|, by WISH 2.) Additionally, we have used the deﬁnition of F to estimate 2 supt∈F |p(t)| . The inequality established immediately above is close to our 1 goal: we need only establish |F | ≥ 4 . To see this, recall that when g(t) = 0, we have p(t) = −q(t). Then WISH 3 guarantees Z Z |p|2 = |q|2 (15) {g=0} {g=0} Z ˆ 2 . |fE| . (16) R\Σ Appealing to Chebyshev’s inequality yields that s Z 1 ˆ 2 |{t ∈ {g = 0}: |p(t)| ≥ |fE| }| ≤ . (17) R\Σ C

1 The ﬁnal step is to show that |{g = 0}| ≥ 2 . But this follows immediately 1 from WISH 1, which says that |E| ≤ 10 , and the observation that |suppg| ≤ |suppf|. Hence 1 1 1 |F | ≥ − ≥ (18) 2 C 4 for C large enough.

2.3 Averaging A key component of Nazarov’s proof of Theorem 4 is a “random periodization” of a given function f. We consider instead functions like gv(t) = P f( k+t ), where v is a random variable uniformly distributed in the in- k∈Z v terval (1, 2). Then following averaging lemma is used to ﬁnd a periodization of f satisfying the wishlist with high probability. Speciﬁcally, it helps us establish WISH 3 above with high probability. WISH 1 can be established with high probability using an even simpler argument. The lemma is applied ˆ 2 to |f| 1R\Σ.

15 Lemma 12. Fix > 0 and suppose φ: R → R is positive and integrable. Then

Z 2 X 1 Z φ(kv)dv ≤ φ (19) 1 06=k∈Z (20)

Proof. We split the sum into ranges of positive and negative k. Then we change variables to obtain

Z 2 X X Z 2k dv φ(kv)dv = φ(u) (21) k 1 k>0 k>0 k 1 Z X dv = φ(u) . (22) k u≥0 k>0: u∈[k,2k]

P 1 But note that k>0: u∈[k,2k] k ≤ 1, so the last display is controlled by 1 Z φ(u)du. (23) u≥0 copying the proof for negative k ﬁnishes the proof.

References

[1] F. Nazarov, Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type . (Russian) Algebra i Analiz 5 (1993), no. 4, 3–66; translation in St. Petersburg Math. J. 5 (1994), no. 4, 663-717.

Michael Bateman, UCLA email: [email protected]

16 3 The endpoint case of a Stein-Tomas theorem for subsets of the real line

after J. Bak and A. Seeger [1] A summary written by Marc Carnovale

Abstract We prove the end point case of a Stein-Tomas theorem for restric- tions to singular sets of high Fourier dimension

3.1 Introduction In 1999, Mockenhaupt [6] showed that the restriction phenomena is much more general than had been previously realized, by demonstrating that Fourier transforms of certain Lp functions may be meaningfully restricted to lie in L2(µ) for µ a (singular) measure of high enough Hausdorff and Fourier dimensions. This is of interest not only because of the deep connections between restriction theorems and a hierarchy of other problems and conjectures, from local smoothing to Kakeya to the Falconer distance set conjecture, with applications to PDE, number theory, geometric measure theory, and, of course, harmonic analysis, (which suggest that anything which sheds light on the traditional question of Fourier restriction is highly significant) - but also because singular sets of given Hausdorff (and, at times, Fourier) dimensions come up, say, in regularity theory of PDE and in Geometric Measure Theory - fields where Harmonic Analysis has already had a powerful impact. And, the result of Mockenhaupt (and later [5]) has seen use, for instance, in [4]. So since the Stein-Tomas theorem includes the end point - that is, they ˆ conclude that kfkL2(dσ) ≤ CkfkLp(Rn) for σ the surface measure on the sphere n−1 2n+2 S and 1 ≤ p ≤ n+3 - it is no surprise that one should like to obtain an end point result for Mockenhaupt’s generalization ˆ kfkL2(dµ) ≤ CkfkLp(Rn) 2(2n − α + β) for1 ≤ p < 4(n − α) + β It is this at-the-time unresolved endpoint case which was the primary focus of Bak and Seeger’s paper. Although Mockenhaupt was able to borrow

17 many of the techniques used by Stein and Tomas, the classical endpoint result follows from embedding the surface measure on the sphere into an (explicit) analytic family and utilizing complex interpolation. This is not obviously available in the general setup of singular measures of given Fourier and Hausdorﬀ dimensions, and so Bak and Seeger were forced to develop an alternative approach. Before describing their contributions, however, let us take a moment to recall the approach to the non-endpoint bounds. Following Tomas, in order to obtain bounds for R |fˆ|2 dµ, it is enough to obtain bounds on the convolution operator f 7→ f ∗µb. Let us record this fact as Lemma 1.

Lemma 1. Suppose that T denotes the operator with kernel µb and that kT kp→p0 ≤ C. Then ˆ 0 kfkL2(dµ) ≤ C kfkp (1) Proof. By applying Plancheral, we have R |fˆ|2 dµ = R f¯(fµˆ )∨ = R f¯(f ∗µˇ) ≤ 2 CkfkpkT fkp0 ≤ Ckfkpkfkp = Ckfkp. As important for us is that the reverse is also true. Proposition 2. Suppose that the bound 1 holds. Then the operator T deﬁned above obeys the bound kT kp→p0 ≤ C. Proof. We have (being somewhat sloppy with our Plancheral and Fourier inversion)

kT fkp0 =

Z 0 1 p 0 ( |f ∗ µb| ) p = Z sup g(f ∗ µb) = kgkp=1 Z sup gˆfdµˆ ≤ kgkp=1 ˆ sup kgˆkL2(dµ)kfkL2(dµ) ≤ kgkp=1

sup kgkpkfkp = kfkp kgkp=1 where we used Cauchy-Schwartz in the penultimate line and the restriction esimate in the ﬁnal line.

18 So nothing is lost by targetting the operator T rather than the restriction estimate directly. In order to obtain bounds on T , we decompose its kernel µb into essentially disjoint frequency regimes by choosing a Schwartz χ0 of −j−1 total mass 1 supported in [−1, 1] with χ| 1 1 ≡ 1, deﬁne χj := χ0(2 ·) − [− 2 , 2 ] −j j j+1 χ0(2 ·) (which is supported in |x| ∈ [2 , 2 ], and deﬁne µj by µj := χjµ P bP b so that µ = j≥0 µj. We may then set Tjf := f ∗ µbj, so that T = Tj. Then the two crucial estimates in [6] and [5] are

jα0 kTjfk1 ≤ C2 kfk∞ (2)

jα1 kTjfk2 ≤ C2 kfk2 (3) where α0 = b is the Fourier dimension of µ and α1 = n − a, with a the Hausdorﬀ dimension of µ. Given our data, these are the best estimates we can obtain for Tj as a mapping between these particular spaces. If we interpolate these bounds using, say, Holder’s inequality, we obtain an esimate 2n−2α+β β k 0 − 0 2n−2α+β 0 p 2 kTjkp→p ≤ C2 , which decays in k for p ≥ 2 β . As we used the best possible estimates in 2 and 3, one might conclude that we won’t get P better than this for kTjkp→p0 . Of course, since T = Tj, we may apply the triangle inequality to conclude an Lp → Lp0 bound on T . This is similar to the case in the Stein-Tomas theorem, where one cannot obtain the endpoint bound through an application of the triangle inequality here - it is the one ineﬃcient point in the proof. Instead, a more clever means of combining the bounds on the pieces Tj of T is necessary, and it is here that the solution introduced by Stein (complex interpolation) has no apparent analogue. This is the context in which Bak and Seeger’s result enters.

3.2 Lions-Peetre Interpolation Spaces One popular (and useful) solution to the abstract question of how to interpolate two (Banach) spaces takes the name of Lions-Peetre interpolation. Suppose that A0 and A1 are two spaces, embeddable in some larger space so that sums of the form a0 + a1, ai ∈ Ai make sense. Suppose further that we have spaces B0 and B1, and a bounded mapping T : Ai → Bi deﬁned on A0 + A1 sending A0 + A1 into B0 ∩ B1. We would like to interpolate the bounds on T between the Ai to bounds on some intermediary space. ¯ The (or rather, a) solution for interpolating between spaces X = (X0,X1) ¯ is to ﬁrst set K(t, x, X) = inf {kxkx0 + tkx1kx1 : x = x0 + x1, xi ∈ xi}. This

19 gives us some weighted combination of the Xi norms - if for instance X1 ⊂ X0 with k · kX0 < k · kX1 , for t large this recovers the X0 norm, while for t = 0 it yields the X1 norm. If K(t, x, X¯) is a type of size estimate up to level t, then the expression R −θ q dt 1 q kxk ¯ := ( (t K(t, x, X¯)) ) q gives a weighted L average of these size Xθ,q t estimates (with the parameter θ controlling the weight). In fact, if we are in an Lp space and replace K(t, x, X¯) by the decreasing rearrangement x∗ of x, then this is precisely the Lorentz norm kxkLθ,q . This fact will take on greater relevance in a moment. ¯ ¯ We call the space endowed with the k · kXθ,q norm, unsurprisingly, Xθ,q. It’s deﬁnition bears some resemblance to that of the Lorentz spaces Lp,q, with norms

Z ∗ 1 q dt 1 kfk p,q = ( (f (t)t p ) ) q ) (4) L t where f ∗ denotes the decreasing rearrangement of f - that is,

∗ f (t) = inf {α ∈ R : λ({x : |f(x)| > α}) ≤ t} (5) (i.e., the function f ∗ has the same distribution of values as f does, but the points which map to these values do so in such a way that f ∗ is monotonic decreasing - morally, f ∗ maps 0 to whatever the most popular value of f is, then the next point to the next most popular value, and so on). Something must also be said for the case q = ∞ - as is natural, we take the supremum 1 ∗ p of the integrand in this case, kfkLp,∞ = supt f (t)t . These Lorentz spaces have the useful properties that Lp,1 is the restricted Lp space (essentially Lp restricted to indicator functions of sets), Lp,p = Lp, and Lp,∞ is weak Lp. They are also useful because they carry more information than Lp norms alone, and so are in many cases the most natural spaces in which to look for optimal bounds on operators. They relate to the Lions-Peetre interpolation spaces in the following manner. ¯ 1 Write the space Xθ,q as [X0,X1]θ,q. Then the following is true: if p = 1−θ θ p0,s p1,s p,q + , then [L ,L ]θ,q = L . p0 p1 ¯ Because when dealing with Lebesgue and Lorentz spaces, Xθ,q is not only concrete but again a Lebesgue or Lorentz space, Lions-Peetre interpolation provides us with useful alternative descriptions of essential spaces. In the

20 following section, we will see one of the ways that this description may be used.

3.3 Skirting the Triangle Inequality via Interpolation The reason we look to Lions-Peetre interpolation is that it affords us a straightforward means of combining two different spaces’ bounds on pieces of an operator to obtain bounds on that operator in an intermediate space, without incurring the losses of a direct application of the triangle inequal- P ity. To make this precise, suppose that an operator T = Tj is defined on A0 + A1, and maps Ai to the space Bi. We assume the bounds

jαi kTjkAi→Bi = Mi2 (6) where α0 < 0 < α1. For a given f that is in A0 and in Ai, these bounds can be combined as

m X X k TjfkB1 + k TjfkB0 ≤ (7) j=0 j>m m X jα1 X jα0 M12 kfkA1 + M02 kfkA0 (8) j=0 j>m for arbitrary positive integers m. This situation is perfectly adapted to the Lions-Peetre interpolation since it is deﬁned in terms of the K functional K(T f, t, B¯), which is a weighted sum of the Bi norms of T f - exactly what we have control over. ¯ To see this, recall that K(T f, t, B) = infg +g =T f kg0kB +tkg1kB . We use P 0 1 0 1 the decomposition of T = Tj to bound this. Let m be a positive integer P Pm to be chosen in a moment, and let g0 = j>m Tjf, g1 = j=0 Tjf. Then as in 7, this is bounded by

m X jα1 X jα0 t M12 kfkA1 + M02 kfkA0 ≤ (9) j=0 j>m

mα1 mα0 C(tM12 kfkA1 + M02 kfkA0 ) (10)

21 We choose m to minimize this - which is achieved when both terms are mα1 mα0 about equal to each other, or tM12 kfkA1 ≈ M02 kfkA0 , which is the m(α0−α1) mα0 −1 same as 2 ≈ tM1kfkA1 (M02 kfkA0 ) . If we set θ = α1 , this minimum is about equal to CM θM 1−θkfkθ kfk1−θ. α1−α0 0 1 A0 A1 Since K is deﬁned via an inﬁmum, we have a uniform bound on K(T f, t, B¯), so we have found that

sup K(T f, t, B¯) ≤ CM θM 1−θkfkθ kfk1−θ (11) 0 1 A0 A1 t>0 Or in other words, we have obtained

kT fk ≤ CM θM 1−θkfkθ kfk1−θ (12) θ,∞ 0 1 A0 A1 The applications of this idea ( ﬁrst appearing in [2] and made explicit in [3]) to the problem at hand are obvious in light of how Lions-Peetre interpolation interacts with Lp and Lp,q spaces. We address these in the next section.

3.4 Finishing up We can now summarize what the ideas in the Bak-Seeger paper are. The ﬁrst has straightforward origins - to obtain an optimal result, one must be careful whenever introducing an inequality to be sure that it is the optimal ˆ one. We have already shown that switching focus from a bound on kfkL2(dµ) to the operator T : g 7→ g ∗ µb is optimal in this sense. And for our data, the bounds on Tj : g 7→ g ∗ µbj in 2 and 3 are the best possible for Tj as mappings from the spaces considered. But what is not optimal is precisely which spaces are being considered in those bounds. So the ﬁrst idea is this - one must replace the optimal bounds within the class of Lp mappings by the optimal Lorentz space bounds. One can get a better estimate on kTjgk2, for jα1 instance, than C2 kgk2 if one is willing to replace the norm on the right by a Lorentz space norm - we obtain strictly stronger information when working with these, and to use any other bounds is to throw away so much that the endpoint becomes unobtainable. The second idea, and perhaps the one which proceeds the above, is to use the Lions-Peetre interpolation to combine the bounds we have available and

22 achieve the optimal Lorentz space estimates. After optimal Lortenz bounds are in hand, these may be interpolated to yield the optimal Lp bounds. To wit, applying the machinery of the previous section to the bounds 2 and 3

jα0 kTjfk1 ≤ C2 kfk∞

jα1 kTjfk2 ≤ C2 kfk2

1 ∞ 1 ∞ (with (A0,A1) = (L ,L ) and (B0,B1) = (L ,L ), α0 = −b, α1 = n−a). α1 ¯ p0,1 1 1−θ θ 2α1−α0 Since θ = , Aθ,1 = L where = + = . Similarly, α1−α0 p0 2 1 2(α1−α0) 0 ¯ p0,∞ Bθ,∞ = L . By 12, this means

kT fk 0 ≤ Ckfk (13) p0,∞ p0

1 2 for any f ∈ A0 ∩ A1 = L ∩ L . Since we are working over R, this isn’t everything - but we can say that it includes indicator functions for sets, and so can concluded the restricted (weak) type estimate kT k 0 . Lp0,17→Lp0,∞ At this point, we can already show (recalling the discussion in Section 3.1) using the bound on T = (g 7→ g ∗ µb) that Z ˆ ¯ ˜ 1 kfkL2(dµ) = ( f(x)f ∗ µb(−x) dx) 2 ≤ (14)

1 ˜ 2 C(kfkLp0,1 kf ∗ µk 0 ≤ (15) b Lp0,∞ Ckfk 0 (16) Lp0,∞ And so we’re done. p0 ,∞ Except we are not done. This is the best restriction estimate from L 0 to L2(dµ). But this bound is not optimal in the sense that we can get a jα1 better norm on the right, because we used the input kTjfk2 ≤ C2 kfk2 which is weaker information than the best Lorentz bound on kTjfk2, which we did not have available. But the restriction estimate that we’ve just shown will actually give us the best Lorentz bound on kTjfk2 - since

23 Z 2 ˆ 2 2 kg ∗ µbjk2 = |f| |µj| ≤ (17) Z ZZ jα1 ˆ 2 jα1 ˆ 2 C2 |f(x)| χˇj ∗ µ(x) dx = C2 |f(x)| χˇj(x − t) dµ(t) dx = (18) Z Z jα1 ˆ 2 C2 χˇj(x)( |f(x + t)| dµ(t)) dx = (19) (20)

The restriction estimate that we have already obtained tells us that 2 the integral over t is bounded by kfkLp0,1 , so the whole is bounded by jα1 R 2 C2 |χˇj|kfkLp0,1 , and by the scaling on χj, this is bounded independent of j. So we replace the naive 3 by the optimal

α0 j 2 kf ∗ µbjk2 ≤ C2 kfkLp0,1 (21) Now we run the argument again using this together with 2. We have 0 d−a eﬀectively replaced our α1 = d − a by α1 = 2 while changing the space A1 from L2 to Lp0,1. The argument for bounding T then gives

kT kLρ,1→Lσ,∞ ≤ C (22) where if one calculates the numbers carefully, they ﬁnd

0 2(α1 − α0)(α1 − α0) ρ = 2 2 and (23) α1 − 3α1α0 + α0 2(α − α0 ) σ = 0 1 (24) α0 Since the operator T is self adjoint, we obtain also the bound

kT kLσ0,1→Lρ0,∞ ≤ C (25) Interpolating between these two bounds gives a range of other optimal Lebesgue and Lorentz space estimates (since interpolating two weak type inequalities yields a strong type, we can move outside of Lp,∞ - in particular, p0 n p ,2 we obtain a bound on T : L 0 (R ) → L 0 ).

24 The restriction estimate that this last is equivalent to is kfkL2(dµ) ≤

Ckfkp0,2. We can leave the Lorentz space once and for all if we interpolate this with the (trivial) optimal L1 → L∞ bound, obtaining

ˆ kfkLq(dµ) ≤ Ckfkp (26)

α1 0 for p ∈ [1, p0] and q = p . α1−α0

References

[1] Bak, J. and Seeger, A., Extensions of the Stein-Tomas theorem. Math. Res. Lett. (2011), no. 18, 767–781;

[2] Bourgain, J., Estimations de certaines fonctions maximales. C. R. Acad. Sci. Paris Ser. I Math. (1985), no.10, 499-502;

[3] Carbery, A. and Seeger, A. and Wainger, S. and Wright, J., Classes of singular integral operators along variable lines. Journal of Geometric Analysis (1999), no. 9, 583-605;

[4] Laba, I., and Pramanik, M., Arithmetic progressions in sets of fractional dimension. Geom. Funct. Anal. (2008), no.2, 429-456;

[5] Mitsis, T., A Stein-Tomas restriction theorem for general measures. Publ. Math. Debrecen (2002), no.1-9, 89-99;

[6] Mockenhaupt, G., Salem sets and restriction properties of Fourier transforms. Geom. Funct. Anal. (2000), no.6, 1579–1587;

Marc Carnovale, UBC email: [email protected]

25 4 Necessary conditions for Lp(Rn)-Fourier multipliers

after V. Lebedev and A. Olevski˘ı[1], [2] A summary written by Vincent Chan

Abstract

We ﬁnd a necessary condition for the indicator function 1E of a n p n measurable set E ⊆ R to be an L (R )-Fourier multiplier for p 6= 2. This is then improved to a necessary condition for a general function f instead of merely indicator functions.

4.1 Introduction Let G be a locally compact Abelian group and let Γ be its dual group. Let m be a bounded, measurable function on Γ, and deﬁne an associated operator T = Tm by Tc f = mfb (1) for f ∈ (Lp ∩ L2)(G), a dense subset of Lp(G) (1 ≤ p ≤ ∞). If T is a bounded operator on Lp(G), then m is called an Lp(G)-multiplier. We denote by Mp(Γ) the space of all these multipliers, and furnish it with the norm

p p kmkMp(Γ) = kT kL (G)→L (G).

Then Mp(Γ) is a Banach space under pointwise multiplication. It is clear that any operator satsfying (1) commutes with translation or is translation- invariant, that is, we have TSx0 = Sx0 T for every translation operator

(Sx0 f)(x) = f(x+x0), x0 ∈ G. It is known that the converse is also true; thus we have an easier way to determine if an operator gives rise to a multiplier. A natural question to ask is now, given a function m ∈ L∞(G) and a ﬁxed p, how do we check if m ∈ Lp(G)? It should be said that the cases p = 1 and p = 2 are already well-understood, but in general this is a diﬃcult question.

4.2 Idempotents case

We will simply things by considering the Γ = Rn and examining only indicator functions.

26 Indicator functions belonging to Mp(Γ) play an important role in several areas of mathematics. For instance, they yield a characterization of p translation invariant, complemented subspaces of L (G). Let Bp(Γ) = {E ⊆ Γ : 1E ∈ Mp(Γ)}; notice this is an algebra of measurable sets. It can be n n n shown that B1(R ) = {∅, R }, and B2(R ) consists of every measurable set. In the case 1 < p < ∞, it is known that Bp(R) contains all intervals, and thus also all ﬁnite unions of intervals. It is natural to ask whether Bp(Γ) is in fact a σ-algebra. In particular, the following question was posed: does there exist a nowhere dense set E of positive measure that belongs to Bp(R) for p 6= 2? Lebedev and Olevski˘ı [1] prove this is not the case; if E has positive measure then it must contain an entire interval. This is a simple consequence of their theorem:

n Theorem 1 (Lebedev, Olevski˘ı). If E ∈ Bp(R ) for p 6= 2 (1 < p < ∞), then E is equivalent to an open set. Here, equivalence means the symmetric diﬀerence of the sets has Lebesgue measure 0; we denote Lebesgue measure by |·|. We use the following notation: B(x, r) is the ball in Rn with centre x and radius r > 0. A point x ∈ Rn is called a density point for E if |E ∩ B(x, r)| lim = 1. r→0+ B(x, r) We use Ed to denote the set of its density points. It is well known that Ed is equivalent to E. For a set E ⊆ Rn, E is its closure and Ec is its complement. We deﬁne the essential boundary of E to be ∂∗E = Ed ∩ (Ec)d. The following can be shown directly: Lemma 2. Both E and Ec are equivalent to open sets if and only if |∂∗E| = 0. The next lemma is the key tool used in the theorem.

∗ Lemma 3. Let |∂ E| > 0. Then for every N ∈ N and any vector {εk = 0, 1}, n 1 ≤ k ≤ N, there exist vectors x0, h ∈ R such that the arithmetic progression d c d xk = x0 + kh (1 ≤ k ≤ N) satisﬁes xk ∈ E if εk = 1 and xk ∈ (E ) if εk = 0.

27 We will prove this lemma by induction, building up the desried vectors (0) (0) one step at a time. Essentially, we begin with vectors x0 , h such that (0) (0) ∗ d x0 +kh ∈ (∂ E) , made possible by the hypothesis. For the inductive step, (j) (j) (j) (j) (j) we assume we have already found vectors x0 , h such that xk = x0 +kh satisfy  Ed if ε = 1, k ≤ j  k (j) c d xk ∈ (E ) if εk = 0, k ≤ j (∂∗E)d if j < k ≤ N. We then make a small perturbation of these vectors so that the new progression will satisfy the same conditions, with one crucial modiﬁcation: the ∗ d d c d (j + 1)st term will be moved from (∂ E) to E if εj+1 = 1 or to (E ) if εj+1 = 0. To do so, we make use of the deﬁnition of denisty points and the essential boundary, and work out the measure theory. To prove the theorem from this lemma will be a proof through contradiction, employing the result of Lemma 2.

4.3 A generalization 12 years later, Lebedev and Olevski˘ıwere able to say much more about these Lp-multipliers, by establishing the essential continuity property for multipliers (again, for the case p 6= 2). We say that a point x ∈ Rn is a point of essential continuity of a function f if, for any ε > 0, there is a neighborhood Ux,ε of x such that |f(y) − f(x)| < ε for almost all y ∈ Ux,ε, and say f is essentially continuous if every point is a point of essential continuity. Lebedev and Olevski˘ı[2] showed:

n Theorem 4 (Lebedev, Olevski˘ı). If f ∈ Mp(R ) for p 6= 2 (1 < p < ∞), then f is almost everywhere essentially continuous.

The proof of this theorem relies on the following lemma, which is a generalization of Lemma 3:

d d Lemma 5. Suppose |E1 ∩ E2 | > 0. Then for every N ∈ N and any vector n {εk = ±1}, 1 ≤ k ≤ N, there exist vectors x0, h ∈ R such that the arithmetic d progression xk = x0 + kh (1 ≤ k ≤ N) satisﬁes xk ∈ E1 if εk = −1 and d xk ∈ E2 if εk = 1.

28 For a complex-valued function f ∈ L∞(Rn), we deﬁne its essential oscillation Ω(f, x) at a point x as

Ω(f, x) = lim ess sup |f(y) − f(x)|. δ→0+ y:|y−x|<δ

Notice if x is a point of essential continuity of f, then Ω(f, x) = 0. To prove the theorem, we assume the set of points for which f is not essentially continuous has positive measure. Then

E = {x ∈ L(f) : Ω(f, x) > ε} has positive measure for some ε > 0, where L(f) is the set of Lebesgue points of f. Let c ∈ R be such that

E1 = {x ∈ E : |f(x) − c| < ε/3} has positive measure, and let

E2 = {x ∈ L(f): |f(x) − c| > 2ε/3} .

d d Since |E1 ∩ E2 | > 0, we can apply Lemma 5 to these sets; the special vector {εk = ±1}, 1 ≤ k ≤ N, will be chosen so that√ the polynomial PN ikx P (x) = k=1 εke satisfies kP kLq(T) ≤ cpkP kL2(T) = cp N. Some inequality manipulations will yield a contradiction as we take N sufficiently large, to complete the proof. There are two things of note at this stage: first, it is easy to check that this is indeed a strengthening of Theorem 1. Secondly, the following is true:

Lemma 6. The following are equivalent:

(a) Almost every point x ∈ Rn is a point of essential continuity of f. (b) There is a function g such that f = g almost everywhere, and g is continuous almost everywhere.

Then with this lemma, the main theorem can be restated as

n Theorem 7 (Lebedev, Olevski˘ı). If f ∈ Mp(R ) for p 6= 2 (1 < p < ∞), then f coincides almost everywhere with a function which is continuous almost everywhere.

29 References

[1] Lebedev, V. and Olevski˘ı,A., Idempotents of Fourier multiplier algebra Geom. Funct. Anal. 4 (1994), pp. 539-544;

[2] Lebedev, V. and Olevski˘ı,A., Fourier Lp-multipliers with bounded powers (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006) pp 129–166; translation in Izv. Math 70 (2006) n0. 3 , 549–585 ;

Vincent Chan, UBC email: [email protected]

30 5 Salem-Bluhm’s construction of Salem sets

after R. Salem [1] and C. Bluhm [2] A summary written by Xianghong Chen

Abstract Given α ∈ (0, 1), we construct a random Cantor set whose Fourier and Hausdorﬀ dimensions equal α almost surely.

5.1 Introduction

1 Recall that in the construction of the standard 3 Cantor set there are three 1 ingredients: the dissection number 2, the dissection ratio 3 and the positions 2 of the subintervals 0 and 3 (which we will call translations). Given α ∈ (0, 1), we are interested in constructing a Cantor-type set whose Fourier dimension (see below for definition) and Hausdorff dimension are both equal to α. Such sets are called Salem sets and were first constructed by Salem [1]. They are special in the sense that they close the gap between the Fourier and Hausdorff dimensions (it is a general fact that the former can not exceed the latter). Salem achieved this by randomizing the dissection ratios and picking in- commensurable translations in the construction of Cantor set. On the other hand one can also instead randomize the translations in order to obtain Salem sets. Both approaches increase the dissection number at each step in order to make up for the ε-loss of decay in the case without such increments. We will follow the second approach which was introduced by Bluhm [2]. In what follows, we will restrict ourselves to R1. All measures are defined on Borel σ-algebra. The Fourier transform of a finite measure µ is defined byµ ˆ(ξ) = R eiξtµ(dt).

5.2 The main result Theorem 1. Given α ∈ (0, 1), there exists a compact set K ⊂ [0, 1] and a probability measure µ supported on K, such that (i) K has Hausdorﬀ dimension α (ii) for all β < α, µˆ(ξ) = O(|ξ|−β/2) as |ξ| → ∞ α (iii) µ(I) . |I| for all interval I.

31 5.3 The set In fact we will construct a class of K most of which will have the properties stated in the theorem. The construction will start with the nominal second step. In the N-th step of the construction, the dissection number will be precisly N, the dissection ratio will be denoted by θN , the translations by XN,j, where j = 1, ··· ,N. − 1 More precisely, for N ∈ N,N ≥ 2, let θN = N α . Notice that

−1 −1 −( 1 −1) −1 N − θN = N (1 − N α ) > cαN .

−( 1 −1) Here we put cα = [1 − 2 α ]/3. Hence cα gives a uniform lower bound for the portion of the gap that an interval of length θN can not ﬁll in an interval of length N −1. j−1 cα j−1 2cα For each N, pick XN,j ∈ [ N + N , N + N ], j = 1, ··· ,N. Then we can correspondingly “dissect” [0, 1] into N disjoint intervals [XN,j,XN,j + θN ].

Now start with N = 2, we “dissect” [0, 1] into two intervals [X2,j2 ,X2,j2 +

θ2], j2 = 1, 2. Then perform the dissection with N = 3 to each [X2,j2 ,X2,j2 +

θ2], we get six intervals [X2,j2 + θ2X3,j3 ,X2,j2 + θ2X3,j3 + θ2θ3], j2 = 1, 2, j3 = 1, 2, 3. Continue the procedure, after the N-th step, we get 2·3 ··· N disjoint closed intervals of the form

[X2,j2 + ··· + θ2 . . . θN−1XN,jN ,X2,j2 + ··· + θ2 . . . θN−1XN,jN + θ2 . . . θN−1θN ].

32 Denote by KN the union of these intervals and set KX = ∩N KN , where the index

N=2,3,··· X = (XN,jN ) jN =1,··· ,N

Then KX is a compact set.

5.4 The measure

Equip KN with the uniform probability measure µN and let FN be its dis- −1 tribution function. Since FN is continuous and kFN − FN+1k∞ ≤ (N!) , FN converges uniformly to a continuous distribution function F . Denote µX = dF , the probability measure corresponding to F , then µX (KX ) = 1.

5.5 The Fourier transform

Since µN converges weakly to µX , in particular,µ ˆN (ξ) → µˆX (ξ), ∀ξ. Notice that for ξ 6= 0,

eiξθ2···θN − 1 1 X iξ(X2,j +···+θ2···θN−1XN,j ) µˆN (ξ) = e 2 N iξθ2 ··· θN N ··· 2 j2,··· ,jN N k iξθ2···θN e − 1 Y 1 X iξθ ···θ X = ( e 2 k−1 k,jk ) iξθ2 ··· θN k k=2 jk=1 Let N → ∞ we get

∞ k Y 1 X iξθ ···θ X µˆ (ξ) = ( e 2 k−1 k,jk ), ∀ξ. X k k=2 jk=1

5.6 Randomization

Now we randomize X such that {XN,jN ,N = 2, 3, ··· , jN = 1, ··· ,N} are jN −1 cα jN −1 2cα independent and each XN,jN is uniformly distributed on [ N + N , N + N ]. In what follows we suppress the subscript X.

5.7 From average decay to deterministic decay It now suﬃces prove the Fourier decay estimate in the average sense. Pre- cisely, for any q, m ∈ N, q, m ≥ 1, we will show that for some constant

33 C = C(α, m, q), 2q −(1− 1 )αq E[|µˆ(ξ)| ] ≤ |ξ| m , ∀|ξ| ≥ C. Assuming this is proven, we can choose q > 2mα−1 and let ξ = n ∈ Z, |n| ≥ C in the above inequality, then

(1− 2 )αq 2q − 1 αq E[|n| m |µˆ(n)| ] ≤ |n| m

Summing over n, we get

X (1− 2 )αq 2q X −2 E[ |n| m |µˆ(n)| ] ≤ |n| < ∞ |n|≥C |n|≥1 Hence −(1− 2 ) α µˆ(n) = O(|n| m 2 ), a.s. In order to pass from the integers to the reals, notice the following Lemma 2 (cf. [3] p.252). Let µ be a probability measure supported on [0, 1] and β > 0 such that µˆ(n) = O(|n|−β), then µˆ(ξ) = O(|ξ|−β). Applying this lemma we see that almost surely we have

−(1− 2 ) α µˆ(ξ) = O(|ξ| m 2 ), ∀m.

5.8 The key estimate To prove the average decay estimate we ﬁrst estimate

k k k 1 X iηX 2q 1 X iη(X +···+X ) X −iη(X +···+X ) E[| e k,j | ] = E[( e k,j1 k,jq )( e k,i1 k,iq )] k k2q j=1 j1,··· ,jq=1 i1,··· ,iq=1 k 1 X X 1 X = E[ 1] + E[ eiη(n1Xk,1+···+nkXk,k)] k2q k2q j1,··· ,jq=1 {i1,··· ,iq} n1,··· ,nk∈Z ={j1,··· ,jq} (n1,··· ,nk)6=0 q! iηnXk,j ≤ q + sup |E[e ]| k j=1,··· ,k n∈Z,n6=0 qq ≤ + 2c−1k|η|−1 kq α

34 5.9 Proof of the average decay

−1 −1 −1 −1 q −q Thus, if 2cα k|η| = 2cα k|ξθ2 ··· θk−1| ≤ q k for k = 2, ··· ,N, then

The above condition holds if and only if it holds for N, or equivalently

1 −1 −q q+1 α 2cα q N [(N − 1)!] ≤ |ξ|

Let N = N(ξ) be maximal such that the inequality is satisﬁed, then N(ξ) is well deﬁned for large |ξ| and is increasing in |ξ| with limit ∞ as |ξ| → ∞. Moreover, due to maximality we have the opposite inequality for N +1. Raise each term to the α-th power we get

αq+α α αq+α cα,qN (N − 1)! ≤ |ξ| ≤ cα,q(N + 1) N!

−1 −q α where cα,q = (2cα q ) . Hence,

1 q N (2 q) αq+α −α 1 N ≤ c (N + 1) |ξ| (2 q q) N! α,q Notice that for N large enough (depending on α, m, q) we have

αq+α 1 N 1 (N + 1) , (2 q q) ≤ [(N − 1)!] 2m

Hence for |ξ| large enough (depending on α, m, q),

35 1 q N 2q 1 (2 q) E[|µˆ(ξ)| ] q ≤ N! −α αq+α 1 N ≤ cα,q|ξ| (N + 1) (2 q q) −α 1 ≤ cα,q|ξ| [(N − 1)!] m

1 α −α − m ≤ cα,q|ξ| cα,q |ξ| m −(1− 1 )α = cα,m,q|ξ| m Since m can be arbitrarily large, one can rid of the constant by choosing larger |ξ|. Raise both sides to the power q, we get for some C = C(α, m, q)

2q −(1− 1 )αq E[|µˆ(ξ)| ] ≤ |ξ| m , ∀|ξ| ≥ C.

5.10 The dimensions

Let K be a compact set in R1, deﬁne the Fourier dimension of K by −β/2 dimF (K) = sup{β ∈ [0, 1] : ∃µ ∈ P(K), s.t.µ ˆ(ξ) = O(|ξ| )} where P(K) denotes the space of probability measures on K. 1 Lemma 3 (cf. [3] p.133). For any compact set K in R , dimF (K) ≤ dimH (K).

Here dimH (K) denotes the Hausdorﬀ dimension of K. Finally, one can show that in the above construction, for any K = KX and µ = µX , dimH (K) = α α and µ(I) . |I| for all interval I.

References

[1] Salem, R., On singular monotonic functions whose spectrum has a given Hausdorﬀ dimension. Ark. Mat. 1 (1951), no. 4, 353–365; [2] Bluhm, C., Random recursive construction of Salem sets. Ark. Mat. 34 (1996), no. 1, 51–63; [3] Kahane, J.-P., Some Random Series of Functions. 2nd ed, 1985, Cam- bridge University Press, Cambridge. Xianghong Chen, University of Wisconsin-Madison email: [email protected]

36 6 Buﬀon’s needle estimates for rational product Cantor sets

after M. Bond, I. Laba, and A. Volberg [2] A summary written by Kyle Hambrook Abstract The probability that Buﬀon’s needle intersects an e−n neighbourhood of a one-dimensional self-similar rational product set, where the factors are deﬁned by at most 6 similarities, is at most Cn−p/ log log n for some p > 0.

6.1 Introduction

The Buﬀon needle probability, or Favard length, of a compact set S ⊂ C is 1 Z π Fav(S) := |projθ(S)|dθ. π 0

Here projθ denotes orthogonal projection onto the line through the ori- iθ0 gin making the angle θ with positive real axis. Pointwise, projθ(re ) := r cos(θ0 − θ). We use |F | to denote the Lebesgue measure of F ⊂ R. A set C is called a self-similar set if there is a positive integer L and distinct, non-collinear points z1, . . . , zL ∈ C such that C is the unique compact SL set for which C = j=1 Tj(C). Here Tj : C → C are the so-called similarity 1 maps defined by Tj(z) = L z + zj. −N We are interested in Fav(SN ), where SN is an L -neighbourhood of a 1 self-similar set S∞. S∞ is of Hausdorff dimension ≤ 1 and of finite H measure. Since the zj are not collinear, S∞ is unrectifiable and the Besicovitch theorem says that |projθ(S∞)| = 0 for almost every θ (see [6]). It follows that

lim Fav(SN ) = Fav(S∞) = 0. (1) N→∞ We are concerned with the rate of decay in (1). The main result of this L paper concerns the rational product set case where {zj}j=1 = A×B for some A, B ⊂ Q. Without loss of generality, we may assume that A, B ⊂ Z and −N min(A) = min(B) = 0. Deﬁne SN = AN × BN + z ∈ C : |z| < L , where −N−1 −N−1 A1 := A, AN+1 := AN + L A, B1 := B, and BN+1 := BN + L B. This is inconsistent with the general deﬁnition of SN that we gave above, but it is equivalent to it up to constants and more convenient to use.

37 −p/ log log N Theorem 1. If SN = AN ×BN and |A|, |B| ≤ 6, then Fav(SN ) . N for some p > 0.

The proof of Theorem 1 is based on a new method of estimating so-called “Riesz products” of trigonometric polynomials. We now outline the proof. The arguments of [7] with the additional modiﬁcations of [3], [4], [5] reduce the proof to the problem of proving lower bounds on integrals of the form

Z 1 n Y j 2 |φt(L ξ)| dξ, (2) −m L j=1 where t = tan(θ) and

1 X φ (ξ) := e2πi(a+tb)ξ. t L (a,b)∈A×B

We describe the reduction in more detail in Section 6.4. The s-th cyclotomic polynomial, for s ∈ N, is

Y 2πid/s Φs(x) := (x − e ). 1≤d≤s (d,s)=1

Q4 (i) Deﬁnition 2. Let F (x) ∈ Z[x]. We write F (x) = i=1 F (x), where each F (i)(x) as a product of irreducible factors of A(x) in Z[x] deﬁned as follows.

(1) Q (1) • F (x) = (1) Φs(x),SF = {s ∈ N :Φs(x) | F (x), (s, L) 6= 1}, s∈SF

(2) Q (2) • F (x) = (2) Φs(x),SF = {s ∈ N :Φs(x) | F (x), (s, L) = 1}, s∈SF • F (3)(x) is the product of those irreducible factors of F (x) that have at 2πiξ0 least one root of the form e , ξ0 ∈ R \ Q, • F (4)(x) is the product of those irreducible factors of F (x) that have no roots on the unit circle.

Deﬁne

F 0(x) := F (1)(x)F (3)(x)F (4)(x),F 00(x) := F (2)(x).

38 We make the following sequence of definitions X 1 A(x) = xa, φ (ξ) = A(e2πiξ), A |A| a∈A X 1 B(x) = xb, φ (ξ) = B(e2πiξ), B |B| b∈B 0 0 2πiξ 0 0 2πiξ 0 0 0 φA(ξ) = A (e ), φB(ξ) = B (e ), φt(ξ) = φA(ξ)φB(tξ), 0 0 2πiξ 00 0 2πiξ 00 00 00 φB(ξ) = B (e ), φB(ξ) = B (e ), φt (ξ) = φA(ξ)φB(tξ). Then 1 1 φ (ξ) = φ0 (ξ)φ00 (ξ), φ (ξ) = φ0 (ξ)φ00 (ξ), A |A| A A B |B| B B 1 φ (ξ) = φ0 (ξ)φ00(ξ) = φ (ξ)φ (tξ). t L t t A B We further define n n n Y j 0 Y 0 j 00 Y 00 j P1(ξ) = φt(L ξ),P1(ξ) = φt(L ξ),P1 (ξ) = φt (L ξ), j=m+1 j=m+1 j=m+1 m m m Y j 0 Y 0 j 00 Y 00 j P2(ξ) = φt(L ξ),P2(ξ) = φt(L ξ),P2 (ξ) = φt (L ξ). j=1 j=1 j=1 Since the integrand in (2) is unchanged under the reflection ξ → −ξ, we can write (2) as

Z 1 n Z Y j 2 1 −m 2 0 2 00 2 |φt(L ξ)| dξ = L |P1(ξ)| |P2(ξ)| |P2 (ξ)| dξ −m 2 −m −m L j=1 [−1,1]\[−L ,L ] Our plan to deduce a lower bound for this integral is as follows. We ﬁrst ﬁnd 0 00 0 00 large subsets V and V of [−1, 1] on which P2 and P2 are large. This leaves us to bound the integral Z 2 |P1(ξ)| dξ. V 0 T V 00\[−L−m,L−m] It is easy to see that this integral is Z Z Z 2 2 2 ≥ |P1| − |P1| − |P1| . (3) V 00 (V 0 S[−L−m,L−m])c [−L−m,L−m]

39 Good upper bounds for the last two integrals are established following the methods of [1], [3], [4], [5], and [7]; Lemmas 10 and 11 below are the precise statements of the upper bounds we use. The method for proving a sufficiently R 2 00 strong lower bound on V 00 |P1| is called Salem’s trick and requires that V have a specific structure. Namely, we will require that V 00 is the support of a function with a non-negative Fourier transform. To meet this requirement, we will choose V 00 to be a difference set Γ − Γ. Indeed, this is why we exploit the symmetry of the integrand with respect to the reflection ξ → −ξ.

0 6.2 The SSV property of φt We now state the terminology and results needed for the choice of V 0.

Deﬁnition 3. Let ϕ : R → C and ψ : N → [0, ∞). We say that ϕ has the SSV property with SSV function ψ if there exist c2, c3 > 0 with c3 c2 such that

( m ) Y k SSV := ξ ∈ [0, 1] : |ϕ(L ξ)| . ψ(m) k=1 is contained in Lc2m intervals of size L−c3m. If ψ(m) equals L−c1m, L−c1m log m, 2 −c1m or L for some c1 > 0, we say that ϕ has the SSV, log-SSV, or square- SSV property, accordingly.

0 In our application, the function ϕ will be either φt or one of its factors 0 Qm 0 k (recall P2(ξ) = k=1 φt(L ξ)), and we will need the constants ci to be uniform in t.

0 2πiξ0 Proposition 4. φA has the log-SSV property. If A(x) has no roots e 0 with ξ0 ∈ R \ Q, then φA has the SSV property. We can arrange for c3/c2 to be as large as we want at the cost of increasing c1. The same assertions hold 0 0 when φA and A(x) are replaced by φB(t · ) and B(x).

Observe that if ϕ1, ϕ2 have the SSV property with SSV function ψ, then so does ϕ1·ϕ2. One consequence of this observation is that we may consider each factor of A separately in proving Proposition 4. The proof that A(1)(e2πiξ) has the SSV property is straightforward argument relying on the fact that Lk Φs(x) and Φs(x ) have common zeroes when (s, L) 6= 1. The proof that A(3)(e2πiξ) has the log-SSV property relies on Baker’s theorem from the theory

40 of Diophantine approximation. Of course, if A(x) has no roots e2πiξ0 with (3) 2πiξ (4) 2πiξ ξ0 ∈ R \ Q, then A (e ) ≡ 1 has the SSV property. As A (e ) is never zero, its contribution to the SSV set can absorbed into constants. The proof As another consequence of our observation, Proposition 4 implies that 0 0 0 φt(ξ) = φA(ξ)φB(tξ) has at least the log-SSV property, and has the SSV 0 0 property if both φA and φB(t·) do. The complement of the set

( m ) Y 0 k SSV (t) := ξ ∈ [0, 1] : |φt(L ξ)| . ψ(m) k=1 (for a particular value of t) will be the set playing the role of V 0.

00 6.3 The SLV structure of φt In this section, we give the terminology and results needed to describe the choice of V 00.

Deﬁnition 5. Let ϕ : R → C. We say that ϕ is SLV-structured if there is a Borel set Γ ⊂ [0, 1] and constants C1,C2 such that

( m ) Y k −C m Γ − Γ ⊂ ξ ∈ R : |ϕ(L ξ)| ≥ L 1 , k=1 −m |Γ| ≥ C2KL

We call Γ an SLV set for ϕ.

00 00 The set V will be Γ − Γ, where Γ is an SLV set for φt . To construct Γ requires that we understand the structure of the set where 00 P2 is large, and to understand this we study which s ∈ N have Φs(x) | A(x) P a (and do similarly for B(x)). Since Φs(x) | A(x) if and only if a∈A ζ = 0 for each primitive s-th root of unity ζ, we are motivated to study so-called linear-multi polygon relations: A ﬁnite set of the form {z1ζ1, . . . , zJ ζJ }, where z1, . . . , zJ ∈ Z and ζ1, . . . , ζJ are roots of unity, is called a linear-multi polygon relation (LMPR) if it satisﬁes

J X zjζj = 0. j=1

41 The proofs of Propositions 6 and 7 below rely on the theory of LMPRs (and, for the latter, a calculation exploiting projective relationships of the form r (ζrs) = ζs). The following proposition implies that we can take Γ = [0, 1] in case |A|= 6 5, |B|= 6 5

Proposition 6. If |A| = 2, 3, 4, 6, then A00(x) ≡ 1. The same holds if A is replaced by B.

The case |A| = |B| = 5 requires signiﬁcantly more eﬀort. It is handled by the combination of the next two propositions.

Proposition 7. Let SA = {r :Φr(x) | A(x), (r, |A|) = 1}. Suppose |A| = 5. There are j0, k0 ∈ N, depending only on A, such that any s ∈ SA has the form j0 k0 s = 2 3 M for some M with (M, 6) = 1. In particular, if s0 = lcm(SA), then for each q | s0 and each q | s0 the set ζa , where ζ is a primitive 2 3 q a∈A q q-th root of unity, is not a LMPR. Analogous assertions hold when A is replaced by B.

Set sA = 1 if {q ∈ N :Φq | A, (q, L) = 1} is empty; otherwise, set sA = lcm({q ∈ N :Φq | A, (q, L) = 1}). Deﬁne sB analogously. Let

ε0 0 N if φt has the SSV property, K = ε0/ log log N 0 N if φt has the log-SSV property where ε0 > 0 is a parameter.

Proposition 8. Suppose that we can write sA = s1,As2,A for integers s1,A, s2,A > 1 such that s2,A < |A| and such that Φq(x) does not divide A(x) for any q | s1,A with (q, L) = 1. Suppose that the analogous thing can be done with B in place of A. Then there is a set Γ that is a ﬁnite union of intervals such that

( m ) Y k −C m Γ − Γ ⊂ ξ ∈ R : |ϕ(L ξ)| ≥ L 1 , k=1 −m |Γ| ≥ C2KL .

00 00 The zeroes of φA and φB are discrete subgroups of R with coarser subgroups removed from them. The idea of Proposition 8 is to arrange for Γ − Γ to be contained in an intersection of neighbourhoods of rescaled copies of

42 these coarser subgroups. This will keep Γ − Γ separated from the zeroes 00 of P2 . In order to ensure that the intersections are generically-sized, hence ensuring that |Γ| is large enough, we use pigeonholing to choose appropriate cosets rather than use groups at each stage.

6.4 Reduction to lower bounds on integrals; required upper bounds In this section, we describe how the proof of Theorem 1 is reduced to estab- lising lower bounds on integrals of the form (2). We also state precisely the upper bounds we need for the last two integrals in (3). We make the change of variable t = tan(θ). This does no harm as we use symmetry to consider only the case θ ∈ [0, π/4]. After rescaling, for z ∈ An × Bn, we may write projθ(z) = a + tb for some a ∈ An, b ∈ Bn. Deﬁne the counting function X −n 2 fn,t := 1projθ(z+[0,L ] ). z∈An×Bn

−n fn,t(x) counts the number of squares (of side L ) that lie “above” or “below” x when the ray forming the angle θ = arctan(t) with the real axis is regarded as the positive “horizontal” direction. Using that supp(fn,t) = projθ(SN ) and self-similarity, we can establish a quantitative version of the statement:

|projθ(SN )| is small if and only if kfn,tk2 is large. Set

n 1 X ν = ∗ ν˜ , ν˜ = δ −k −k . n k=1 k k L L a+tL b (a,b)∈A×B Then n n ˆ n Y −k fn,t = L 1[0,L−n] ∗ νn, fn,t(ξ) = L 1ˆ[0,L−n](ξ) · φt(L ξ). k=1 ˆ Since kfn,tk2 = kfn,tk2, and since we can use a pigeonholing argument to eﬀec- n tively neglect the decay factor L 1b[0,L−n](ξ), the task of bounding |projθ(SN )| from above is reduced to the problem of proving a lower bound on Qn −k k k=1 φt(L ξ)k2. The following proposition states the reduction precisely; the set E appearing below is a set of directions t = tan(θ) which we do not deﬁne here.

43 Proposition 9. Theorem 1.2 is implied by the following statement: If ε0 > 0 is suﬃciently small, and if |E| ≥ 1/2K1/2, there is a t ∈ E such that

Z 1 n Y j 2 −n −αε0 |φt(L ξ)| dx ≥ cKL N . −m L j=1

The following two lemmas furnish the upper bounds we need on the last two integrals in (3).

Lemma 10. For t ∈ E, we have

Z L−m 2 −n |P1| ≤ 4C0KL . (4) −L−m

Lemma 11. Assume φt has the log-SSV property. If ε0 > 0 is suﬃciently 1/2 small, and if |E| ≥ 1/2K , there exists a t0 ∈ E such that Z 2 −n |P1,t0 (ξ)| dξ ≤ C0KL . T −1 SSV(t0) [L ,1]

6.5 The main argument Now we prove the statement which, according to Proposition 9, implies The- orem 1.

1/2 Proposition 12. If ε0 > 0 is suﬃciently small, and if |E| ≥ 1/2K , there is a t ∈ E such that

Z 1 n Y j 2 −n −αε0 |φt(L ξ)| dξ ≥ cKL N . (5) −m L j=1

Proof. By Proposition 4, φt has at least the log-SSV property. Assume |E| ≥ 1/2 1/2K , and assume ε0 > 0 is small enough that the hypothesis of Lemma 11 is satisﬁed. Let t = t0 be the direction in E that Lemma 11 furnishes. Write the intergral in (5) as Z 1 −m 2 0 2 00 2 L |P1(ξ)| |P2(ξ)| |P2 (ξ)| dξ. (6) 2 [−1,1]\[−L−m,L−m]

44 0 00 According to Section 6.2, we can find a SLV set Γ for P2. So, since |P2 | ≥ L−C1m on Γ − Γ, (6) is 1 Z −2C1m−m 2 0 2 ≥ L |P1| |P2| 2 Γ−Γ\[−L−m,L−m] 0 c By the definition of SSV (t0), |P2| & ψ(m) on SSV(t0) . Hence (6) is Z −2C1m−m 2 2 & L ψ(m) |P1| −m −m T c (Γ−Γ\[−L ,L ]) SSV(t0) Z Z Z −2C1m−m 2 2 2 2 ≥ L ψ(m) |P1| − |P1| − |P1| . −m m −m T Γ−Γ [−L ,L ] [L ,1] SSV(t0) R 2 To bound Γ−Γ |P1| , we employ Salem’s trick on difference sets. Write −n−m P 2πiαξ n−m −1 P1(ξ) = L α∈A e , and note |A| = L . Let h = |Γ| 1Γ ∗ 1−Γ. ˆ −1 2 Then 0 ≤ h ≤ 1 and h = |Γ| |1cΓ| ≥ 0. Hence Z Z 2 2 |P1(ξ)| dξ ≥ |P1(ξ)| h(ξ)dξ Γ−Γ Γ−Γ Z X 0 = L−2(n−m) h(ξ)e2πi(α−α )ξdξ α,α0∈A Γ−Γ X = L−2(n−m) hˆ(α − α0) α,α0∈A X ≥ L−2(n−m) hˆ(0) = L−2(n−m)|Γ| · |A| α∈A −m −(n−m) −n ≥ C2KL L = C2KL .

R 2 Since C2 > 5C0, by combining the bound for Γ−Γ |P1| with Lemmas 10 and 11 we ﬁnd that (6) is

−2C m−m 2 −n −n −αε & L 1 ψ(m) KL & KL N 0 for some α > 0. The last inequality is true by the SSV or log-SSV property 0 of φt and an appropriate choice of m

References

[1] Bond, M., Combinatorial and Fourier Analytic L2 Methods For Buﬀon’s Needle Problem.

45 http://bondmatt.wordpress.com/2011/03/02/thesissecond-complete- draft/

[2] Bond, M., Laba, I., and Volberg, A., Buﬀon’s needle estimates for rational product Cantor sets.

[3] Bond, M. and Volberg, A., Buﬀon needle lands in ε-neighbourhoood of a 1-dimensional Sierpinski Gasket with probability at most | log ε|−c. Comptes Rendus Mathematique, 348 (2010), Issues 11-12, 653-656.

[4] Bond, M. and Volberg, A., Buﬀon’s needle landing near Besicovitch irregular self-similar sets. http://arxiv.org/abs/0912.5111

[5] Laba, I. and Zhai, K., The Favard length of product Cantor sets. Bull. London Math. Soc., 42 (2010), 370-377.

[6] Mattila, P., Geometry of Sets and Measures in Euclidean Spaces. Cam- bridge University Press, 1995.

[7] Nazarov, I., Peres, Y., and Volberg, A., The power law for the Buﬀon needle probability of the four-corner Cantor set. Algebra i Analiz, 22 (2010), 82-97; translation in St. Petersburg Math. J., 22 (2010), 6172.

Kyle Hambrook, UBC email: [email protected]

46 7 Projecting the One-Dimensional Sierpinski Gasket

after Richard Kenyon [3] A summary written by Edward Kroc

Abstract We deﬁne the so-called one-dimensional Sierpinski gasket, calcu- late the measure of its linear projections in any direction and obtain bounds on the Hausdorﬀ dimension of these projections.

7.1 Introduction Deﬁne

( ∞ ) X −i S = αi3 | αi ∈ {(0, 0), (1, 0), (0, 1)} . i=1 It is easy to see that S may be described as the set

S = {(x, y) ∈ C × C | x + y ∈ C} where C is the ordinary “middle third” Cantor set constructed on [0, 1/2]. 2 x y Equivalently, S is the attractor in R for the three linear maps (x, y) 7→ ( 3 , 3 ), x+1 y x y+1 (x, y) 7→ ( 3 , 3 ), (x, y) 7→ ( 3 , 3 ). The set S resembles the Sierpinski gasket, obtained by replacing all occurrences of 3 with 2 in the first (or third) description. We call S the one- dimensional Sierpenski gasket since S is a union of three copies of itself, each scaled by a factor of 1/3, and thus has Hausdorff dimension 1. Our main objects of study are the various linear projections of S onto the x-axis: define Su = πu(S), where

1 u π = . u 0 0

Notice that S0 = C and S1/2 = [0, 1/2]. In general, Su is the set of real numbers which have an expansion using negative powers of 3 and digits {0, 1, u}. Note that diam(Su) = 1/2 if 0 1, and diam(Su) = (1 + u)/2 if u < 0.

47 We will be concerned with both the linear measure and the dimension of the various projections Su. Througout this summary, we let µ denote one- dimensional Lebesgue measure and dim(E) denote the Hausdorﬀ dimension of the set E. A theorem of Besicovitch [1] states that the projection in almost every direction of an irregular set of Hausdorﬀ dimension 1 in R2 has Lebesgue one-dimensional measure 0. We compute the measure of Su for every u.

Theorem 1. µ(Su) > 0 iﬀ u is a rational of the form p/q in lowest terms with p + q ≡ 0 mod 3. In this case, µ(Su) = 1/q. This result gives a simple criterion for deciding when a set of 3 nonnegative numbers represents a set of positive measure in base 3, a question formulated by Odlyzko [5]. Concerning the dimension of our projections, a theorem of Marstrand [4] states that almost every linear projection of a set of dimension 1 in R2 has dimension 1. Furstenberg has conjectured that dim(Su) = 1 for every irrational u; this remains open. We obtain bounds for the dimension of Su when u is close to a single rational (or is well-approximated by an appropriate sequence of rationals). A uniform bound on dim(Su) of 1−log(5/3)/2 log 3 > 0.767 for any irrational u has since been obtained by Swi¸atekand´ Veerman [6] using energy estimates for certain natural measures supported on the projections Su.

7.2 The measure of the projections We prove the forward direction of Theorem 1 via a chain of lemmas. The opposite direction follows from our lemmas and the fact that dim(Su) < 1 if u = p/q with p + q 6≡ 0 mod 3 (see [3]).

Lemma 2. If µ(Su) > 0, then Su contains an interval. Sketch of proof. By deﬁnition,

( ∞ ) X −i Su = ai3 | ai ∈ {0, 1, u} ; i=1 thus, 3Su = Su ∪ (Su + 1) ∪ (Su + u), (1)

48 where for A ⊂ R and x ∈ R, xA = {xa | a ∈ A} and x+A = {x+a | a ∈ A}. By subadditivity and the scaling property of Lebesgue measure, we see that the three translated copies of Su which cover 3Su are disjoint in measure. n n Similarly, for any n ≥ 0, 3 Su is covered by at most 3 translates of Su pairwise disjoint in measure. Since µ(Su) > 0, we may arrange these translates to “ﬁll out” any interval that contains a Lebesgue point of Su. Since Su is closed, we invoke the Baire category theorem to conclude that Su contains an interval.

Lemma 3. If Su contains an interval, then u is rational. Sketch of proof. This can be seen by appealing to the self-similarity of the set Su. We may tile the real line in a periodic manner using the translates of Su n from 3 Su (we call these translates “tiles”). The possibility of such a tiling follows from the fact that if Su has interior, then it is the closure of its interior and that the boundary of Su has measure 0. Periodicity is imposed on any such tiling of R by virtue of the invariance of the tiling under expansion (by a factor of 3) and subdivision of the tiles (as in (1)). A simple calculation shows that such periodicity is only possible if u is rational.

Lemma 4. If u = p/q in lowest terms with p + q ≡ 0 mod 3, then µ(Su) = 1/q. n Pn −i Sketch of proof. Let 0 < p < q. Define Sp,q = { i=1 ai3 | ai ∈ {0, p, q}}, so ∞ n n that Sp,q = qSu. Now Sp,q consists of 3 triadic rationals which are distinct since p + q ≡ 0 mod 3. −n P Define probability measures µn = 3 n δ(x) where δ(x) is the unit x∈Sp,q point mass at x. For each triadic interval I = [p3−k, (p + 1)3−k) with k ≤ n ∞ we have µn(I) ≤ µ(I). Since Sp,q is closed, we take a weak limit µ∞ of µn to ∞ ∞ find that 1 = µ∞(Sp,q) ≤ µ(Sp,q). ∞ We tile R as in the proof of Lemma 3 using translates of Sp,q. Showing ∞ that the period of this tiling is 1 will yield µ(Sp,q) = 1, and thus µ(Su) = 1/q. Let R ∈ Z+ be a period of the tiling. Then there is a set W ⊆ [0,R) ∩ Z ∞ such that each tile is of the form x + Sp,q, where x ∈ W + RZ, and there are ∞ tiles at each point of W + RZ. By the self-similarity of Sp,q, the set W taken modulo R is invariant under the three maps x 7→ 3x, x 7→ 3x+p, x 7→ 3x+q. Let G be the directed graph with vertices V = [0,R) ∩ Z and edges from x to (3x + d) mod R for each d ∈ {0, p, q}. Let f be an eigenvector for the adjacency matrix T with eigenvalue 3: T f(x) = f(3x) + f(3x + p) + f(3x + q) = 3f(x), (2)

49 where the arithmetic in the arguments is done mod R. Note that we may always choose R large enough so that T has an eigenvalue of 3. Assume that 3 - R (the alternative case is similar). Identifying V with ZR, we write the nth Fourier coeﬃcient of 3φ(R)f and simplify using (2). This gives that the nth Fourier coeﬃcient is nonzero only if n = 0. Consequently, f is constant on W (its support) and so W = [0,R) ∩ Z.

7.3 Bounds on the dimension of the projections

1 k k Pk −i If u is rational, dim(Su) = lim log3 |Su|, where Su = { i=1 ai3 | ai ∈ k→∞ k {0, 1, u}}; i.e. dim(Su) equals the Minkowski dimension of Su. This follows from a counting argument and a theorem of Falconer [2] that guarantees the Minkowski and Hausdorﬀ dimensions of self-similar sets agree.

Theorem 5. Let u be a real number and pi/qi a sequence of rationals such that pi+qi ≡ 0 mod 3, qi → ∞, and such that there exists constants C, α > 0 for which

pi C u − < α . qi qi

Then dim(Su) ≥ 1 − 1/α. In particular, if u is an appropriate Liouville number, then dim(Su) = 1. This is a residual subset of R. −k k Sketch of proof. Suppose |u−p/q| < (2/3)3 /q. The set Sp/q ⊂ Sp/q consists of 3k distinct points on the lattice (3−k/q)Z. So we require at least 3k intervals −k of length 3 /q to cover Sp/q. 0 For each x ∈ Sp/q, let x be the point with the same sequence of digits, replacing all occurrences of the digit p/q with u. Then

∞ −k 0 X p −i 3 p 3 |x − x | < u − 3 = u − < . q 2 q q i=1 k −k Thus it takes at least (1/3)3 intervals of length 3 /q to cover Su. Let Nu() be the minimum number of intervals of length needed to cover −k α Su. Setting 3 /qi = C/qi , we estimate the Minkowski dimension of Su as α α−1 log Nu(C/qi ) log(qi ) + C 1 dimM (Su) = lim ≥ lim ≥ 1 − . i→∞ α log qi + C i→∞ α log qi + C α We again invoke the theorem of Falconer [2] to conclude the same bound on dim(Su).

50 When u is close to a single rational, the previous argument can be reﬁned to give a lower bound on dim(Su). Let p/q be in lowest terms with p + q ≡ 0 mod 3 and 0 < p < q. For a N sequence {ai} ∈ {0, p, q} , deﬁne

∞ X −i r({a1, a2,...}) = ai3 ∈ qSp/q. i=1

We build a graph G∂ to describe sequences {ai} for which r({ai}) is in q−1 the boundary ∂S of S. Assume q > 2 and let m = b 2 c. The graph G∂ has 2m vertices, labelled with nonzero integers from −m to m inclusive. From a vertex labelled x, for every d1, d2 ∈ {0, p, q} there is an edge labelled (d1, d2) pointing to vertex 3x + d1 − d2 if this is a nonzero integer in [−m, m]. For each vertex v of G∂ there is a word of length < log3 q which does not label a path from v (to see this: if v > 0, take a word consisting entirely of qs; otherwise, take a word consisting entirely of 0s). Concatenating these words appropriately, we construct a word γ of length c < q log3 q which does not label a path starting at any vertex of G∂. Theorem 6. Let u ∈ and suppose there are relatively prime integers p, q, R 0 1 − log 1/ . q log q − 1

N Sketch of proof. For each k > 0, let Wk ⊂ {0, p, q} be the set of sequences j q −k a = {ai} such that for each j ≥ 0, r(σ a) is at a distance at least 2 3 from ∂S, where σ is the left shift. An element of Wk is then a sequence such that no substring of length k labels a path in G∂. When k is suﬃciently large, we claim that the growth rate of Wk is close to 3 and approximates qSp/q from below. Consequently, we see that dim(Sp/q) is close to 1, and replacing occurrences of p/q by u as in the previous proof will lead to a similar bound on dim(Su). Let k > 2c. Then Wk contains all sequences of the form γ, w1, γ, w2, γ, . . ., (N) where the wj are arbitrary words of length k−2c. Let Wk denote the subset (N) of N-truncations of elements of Wk. If Wk contains l arbitrary words of length k − 2c, then N ≈ l(c + k − 2c). So it takes about 3l(k−2c) many words

51 (N) of length l(k − c) to cover Wk ; thus, the growth rate of Wk is at least k−2c q log q ξ = 3 k−c . Since c < q log q, we have log ξ > 1 − 3 . 3 3 k−q log3 q 0 Now we let Wk be the set of sequences obtained from Wk by replacing all occurrences of digit p by the digit qu. By hypothesis, any correspondent 0 0 q sequences a ∈ Wk and a ∈ Wk lie within 2 of each other, where k = 0 0 0 0 0 q −k −blog3 3c. By deﬁnition of Wk, if a , b ∈ Wk, then |r(a ) − r(b )| ≥ 6 3 . 0(n) n q −n So for n > k, it takes at least #(Wk ) ≈ ξ many intervals of length 6 3 0 to cover Wk. Thus, the Minkowski dimension of Su (and so the Hausdorﬀ dimension by [2]) is at least log3 ξ.

References

[1] Besicovitch, A. S., On the fundamental geometric properties of linearly measurable plane sets of points III, Mathematische Annalen 116 (1939), 349–357.

[2] Falconer, K. J., Dimensions and measures of quasi self-similar sets, Proc. American Math. Soc. 106 (1989), 543–554.

[3] Kenyon, R., Projecting the one-dimensional Sierpinski gasket, Israel Jour. of Math. 97 (1997), 221–238.

[4] Marstrand, J., Some fundamental geometric properties of plane sets of fractional dimension, Proc. London Math. Soc. 4 (1954), 257–302.

[5] Odlyzko, A., Nonnegative digit sets in positional number systems, Proc. London Math. Soc. 37 (1978) 213–229.

[6] Swi¸atek,G.,´ Veerman, J. J. P., On a conjecture of Furstenberg, Israel Jour. of Math. 130 (2002), 145–155.

Edward Kroc, University of British Columbia email: [email protected]

52 8 Wiener’s ‘closure of translates’ problem and Piatetski-Shapiro’s uniqueness phenomenon

after N. Lev and A. Olevskii [2] A summary written by Allison Lewko

Abstract We present a counterexample to a conjecture of N. Wiener, showing p p that the cyclic vectors in ` (Z) (and L (R)) cannot be characterized in terms of the zero set of the Fourier transform for values of p in the range 1 < p < 2.

8.1 Introduction

In [4], Wiener characterized the cyclic vectors in `p(Z) and Lp(R) for p = 1, 2 in terms of the zeros of the Fourier transform. We will focus on `p(Z) for p simplicity, as the situation in L (R) is similar. A vector c = {cn}n∈Z is called a cyclic vector in `p(Z) (with respect to translations) if the linear span of its translates is dense in `p(Z). Wiener proved:

Theorem 1. [4] Let c = {cn}n∈Z. (i) c is a cyclic vector in `2(Z) if and only if the Fourier transform

X int cˆ(t) := cne n∈Z is nonzero almost everywhere. (ii) c is cyclic in `1(Z) if and only if cˆ(t) has no zeros. One can interpret these results as being uniﬁed by the feature that c is a cyclic vector if and only if the set

Zcˆ := {t ∈ T : cˆ(t) = 0} containing the zeros of the Fourier transform is suﬃciently “small,” where the appropriate notion of smallness depends on whether p = 1 or p = 2. Wiener expected that the intermediate range of 1 < p < 2 would behave like some interpolation of these two results, with cyclicity being characterized entirely by Zcˆ. It is quite natural to guess that one would need an intermediate

53 notion of “smallness” for Zcˆ parameterized by p, as it is known for 1 < p < 2 that requiring Zcˆ = ∅ is too strong and requiring Zcˆ have Lebesgue measure zero is too weak. One positive result along these lines was proved by A. Beurling [1], who showed that when the Hausdorﬀ dimension of Zcˆ is suﬃciently small with respect to p, then c is a cyclic vector in `p(Z) for 1 < p < 2. Despite being sharp, this criterion is not necessary, and hence provides only an implication and not a characterization. The main result of this work is that for any 1 < p < 2, one cannot p characterize the cyclic vectors in ` (Z) by any criteria depending only on Zcˆ. This is a corollary of the following theorem:

Theorem 2. [2] Let 1 0, and cˆ vanishes on K, then c is not cyclic in `p(Z). (b) There exists c ∈ `1(Z) such that cˆ vanishes on K, and c is a cyclic vector in `p(Z). This yields:

Corollary 3. [2] Given any p, 1 < p < 2, one can ﬁnd two vectors in `1(Z) such that one is cyclic in `p(Z) and the other is not, but their Fourier transforms have an identical set of zeros.

We thus have a deﬁnitive counterexample to Wiener’s conjecture for these intermediate values of p. The methods for obtaining the proof of the above theorem are inspired by the Piatetski-Shapiro phenomenon. This phenomenon refers to compact sets in T representing a surprising balance of “smallness” and “largeness” properties. More precisely, Piatetski-Shapiro [3] constructed a compact set K supporting a nonzero distribution S with Fourier transform Sˆ(n) tending to zero as |n| → ∞, but not supporting any such measure. This result was unexpected, as it was previously believed that any compact set supporting such a distribution would also support such a measure. The connection between the Piatetski-Shapiro phenomenon and the characterization of cyclic vectors is expressed by the following known facts. First, p if c is a non-cyclic vector in ` (Z), then Zcˆ supports a nonzero distribution with Fourier coeﬃcients in `q(Z), where q := p/(p − 1). The converse of this

54 statement turns out to be false, but two closely related results are known. q If Zcˆ supports a nonzero measure with Fourier coefficients in ` (Z), then c is a noncyclic vector in `p(Z). Also, if cˆ is continuously differentiable and p Zcˆ supports such a nonzero distribution, then c is noncyclic in ` (Z). Thus, the strategy for proving the main theorem is to find a compact set K that is “large enough” to support such a nonzero distribution but “small enough” so that it does not support such a measure.

8.2 Tools We now give a high level outline of the proof. We first define an appropriate notion of smallness for K which suffices to rule out the support of measures with quickly vanishing Fourier coefficients while still allowing us to build a suitable distribution S supported on K. We employ the notion of a Helson set:

Definition 4. A compact set K is called a Helson set if it satisfies any one of the following equivalent conditions: (i) Every continuous function on K admits extension to a function whose Fourier coefficients are in `1(Z). (ii) There exists a constant δ1(K) > 0 such that, for every measure µ supported by K, Z sup |µˆ(n)| ≥ δ1(K) |dµ|. n∈Z

(iii) There exists a constant δ2(K) > 0 such that, for every measure µ supported by K, Z lim sup |µˆ(n)| ≥ δ2(K) |dµ|. |n|→∞ To establish that any Helson set is suﬃciently “small” for our purposes, we must show every Helson set is contained in Zcˆ for some cyclic c. For this, we prove that for any Helson set K, the set of vectors c with Zcˆ ⊇ K that are also cyclic in `p(Z) for every p > 1 is a countable intersection of open, dense sets. Hence, by the Baire category theorem, this set is nonempty. In order to construct a Helson set supporting a suitable distribution, it helps to identify some alternate properties that imply that a set K is a Helson set. In particular, we consider totally disconnected compact sets K which, given any real-valued continuous function h on T with no zeros in K, allow

55 us to find a real trigonometric polynomial P which has absolute value > 1 on K, agrees with the sign of h everywhere on K, and has Fourier coefficients bounded independently of h in terms of the `1-norm. We prove that any such set K is also a Helson set. This alternate set of criteria enables us to take an iterative approach to constructing a Helson set K and the suitable distribution S it supports. The main idea is to first construct a compact set K with much weaker properties. More precisely, it should support a reasonable approximation to the constant 1 function (close in terms of the `q-norm on the Fourier coefficients) while also allowing one to find a trigonometric polynomial P with the properties described above with respect to a specific, fixed h which is also a trigonometric polynomial. Once we find such sets K1,K2,... for a dense sequence of trigonometric polynomials, then we can take their intersection to arrive at a final set K which is then a Helson set. We can also take the infinite product of the supported approximations to the constant 1 function as our nonzero supported distribution with sufficiently vanishing Fourier coefficients. This is merely a birds-eye view of the outer structure of the proof - there are many technical challenges arising in the implementation, particularly in constructing the initial compact sets and 1-approximations to combine.

References

[1] Beurling, A., On a closure problem. Ark. Mat. 1 (1951), 301–303. [2] Lev, N. and Olevskii, A., Wiener’s ‘closure of translates’ problem and Piatetski-Shapiro’s uniqueness phenomonon. Ann. of Math. 174 (2011), 519–541; [3] Pyatecki˘i-Sapiro,ˇ I. I., Supplement to the work “On the problem of uniqueness of expansion of a function in a trigonometric series.” Moskov. Gos. Univ. Uˇc.Zap. Mat. 165(7) (1954), 79–97, English translation in Selected Works of Ilya Piatetski-Shapiro, Amer. Math. Soc. Collected Works 15, 2000; [4] Wiener, N., Tauberian theorems. Ann. of Math. 33 (1932), 1–100; Allison Lewko, University of Texas at Austin and Microsoft Research New England email: [email protected]

56 9 Bounded orthogonality systems and the Λ(p)- set problem II

after Jean Bourgain [1] A summary written by Mark Lewko

9.1 Introduction Here we outline the proof of the following theorem [1] (for motivation and background we refer the reader to Stefan’s summary):

N Theorem 1. Let {φn(x)}n=1 be a bounded orthogonal system and 2 n2/p such that X || anφn(x)||p p ||{a}||`2 n∈S

Let ξn(ω) denote independent selectors taking the value 1 with probability 2/p−1 N δ = n and 0 otherwise. We will let A := {an}n=1 denote a sequence of real numbers. The proof will proceed by estimating the quantity

N X p E sup || anξnφn(x)||p. (1) ||{a}||`2 ≤1 n=1 Ultimately, one can show that this quantity is O(1). On the other hand, standard deviation inequalities show that the size of the selected set, S(ω) := 2/p {n ∈ [N], ξn(ω) = 1)}, will be within a constant factor of n with all but exponentially small probability. These two facts easily imply Theorem 1. Expanding the norm in (1), this may be rewritten as

* N N N p−2+

X X X E sup anξn(ω)φn(x), anξn(ω)φn(x) anξn(ω)φn(x) (2) ||a||`2 n=1 n=1 n=1 We will now make a simplifying assumption: we will restrict attention to the case when all of the coefficients are of the same scale (that is we restrict to −i a coefficient level set). In other words we consider B ⊂ [N] such that an ∼ 2 for n ∈ B (here we use a ∼ b to denote b/2 ≤ a ≤ b). Clearly we have |B| = O(22i). A considerable amount of the technical difficulties of Bourgain’s proof

57 stem from handling all possible level sets simultaneously. Roughly speaking the idea is to prove several (stronger) ‘restricted’ multilinear variants of the level sets estimates and then interpolate between them to recover the full result. (Some of these ideas are discussed in Stefan’s summary.) Of course, from dyadic pigeonholing one can use this single level set (or ‘restricted’) result to obtain the general result with a loss of log(N) in the Λ(p) constant. Let us now deﬁne the random variable

N X p Kp(ω) := sup sup || anξn(ω)φn(x)||p. B ||{a}||`2 ≤1 n=1 −1/2 ai∼|B|

Our goal is to show that EKp(ω) = O(1) . Expanding the norm (and omitting the sup over B for brevity) we have

N 1 X E sup sup p 1B(n)ξn(ω) ||{a}||`2 B⊆[N] |B| n=1 −1/2 ai∼|B|

* N N p−2+ X X × φn(x), anξn(ω)φn(x) anξn(ω)φn(x) . (3)

n=1 n=1 This expression turns out to be difficult to work with because of joint dependencies on ω between the first and second coordinate of the inner product. The first key idea is that one may ‘decouple’ the dependencies using some fairly elementary probabilistic inequalities. The upshot is that the above quantity can be essentially controlled by

N 1 X sup sup 1 (n)ξ (ω ) Eω1 Eω2 p B n 1 ||{a}||`2 B⊆[N] |B| n=1 −1/2 ai∼|B|

* N N p−2+ X X × φn(x), anξn(ω2)φn(x) anξn(ω2)φn(x) . (4)

n=1 n=1

Here ω1 and ω2 are independent. Now, for ﬁxed ω2, we may deﬁne

58 N Q(ω2) ⊂ R by  * N N p−2+  X X Q(ω2) =  φm(x), anξn(ω)φn(x) anξn(ω)φn(x)  :  n=1 n=1 1≤n≤N ) {a} ∈ `2, supp{a} ≤ |B| .

Thus our quantity can now be rewritten as X Eω2 Eω1 sup ξi(ω1)yi. (5) y∈Q(ω2) i∈B

If we replaced the random selectors ξi(ω1) with independent Gaussians then the above quantity is classically studied under the heading of ‘boud- edness of Gaussian processes’. A central result in this theory is Dudley’s entropy bound which states that the (Gaussian variant) of the above quantity is controlled by the entropy (or covering) numbers of the set Q(ω2). It turns out that Dudley’s method can be adapted to the case of selector processes as well. n q Let Q ⊂ R and let Nq(Q, t) denote the number of ` balls of radius t needed to cover Q. We then have

Lemma 2. (Chaining Lemma) Let ξi denote independent selectors of mean δ. Then, denoting the diameter of Q as diam(Q), we have

X p −1 E sup ξi(ω)yi δm + log(δ )Diam(Q) y∈Q i∈A |A|=m

Z ∞ −1/2 −1 1/2 + log (δ ) [log N2(Q, t)] dt. 0 In fact, slightly stronger bounds are available (and required in the arguments needed to remove the level set restriction) however we omit this for brevity. We have now reduced matters to understanding the entropy function P N2(Q(ω2), t) and the quantity diam(Q(ω2)). Denote P|B| := { i∈A aiφi(x): N |A| ≤ |B|}. Let f, g ∈ P|B| and vf , wg ∈ Q(ω2) the associated vectors in R . We then have

59 N ! X p−2 p−2 2 p−2 p−2 |vf − wg| ≤ | φi, f|f| | − | φi, g|g| | ≤ ||f|f| − g|g| ||2 i=1 by Bessel’s inequality. We now make the further assumption that 2 < p < 3 (the case 3 ≤ p < 4 can be handled by a similar argument. The result for larger p can subsequently be obtained by a more complicated iterative version of these arguments. We omit discussion of this here). Invoking the pointwise bound (using 2 < p < 3) f|f|p−2 −g|g|p−2 |f −g|(|f|p−2 +|g|p−2) we may further bound the above by

p−2 p−2 ≤ ||(f − g)|f| ||2 + ||(f − g)|g| ||2 ≤ ||f − g|| 2 3−p where the last inequality follows from Holder’s inequality and the fact that ||f||2, ||g||2 ≤ 1. From this, we see that N2(Q(ω2), t) N p−2 (P|B|, t/2). In 4 other words the quantity N2(Q(ω2), t) can be controlled by the entropy numbers N p−2 (P|B|, t/2). A similar calculation using Bessel’s inequality shows 4 that

p/2 p−2 diam(Q(ω2)) ≤ Kp(ω2) |B| 4 .

It now suﬃces to gain control of the entropy numbers N p−2 (P|B|, t/2). 4 This is the content of the following lemma.

Lemma 3. (Entropy lemma) For t < 1

N log N (P , ct) |B| log 1 + + log(1/t) q |B| |B| and for t ≥ 1 N log N (P , ct) |B| log 1 + t−ν q |B| |B| for some ν := ν(q) > 2.

This is proven using what has become known as the ‘support reduction trick’ and Khinchin’s inequality. Collecting the estimates we have that (5) can be bounded by

p−2 1/2 4 p/2 −1/2 −1/2 −1 δ |B| Eω2 Kp (ω2) + |B| log (δ )

60 N Z 1 Z ∞ ×|B|1/2 log1/2(1 + ) log1/2(1/t)dt + t−ν/2dt |B| 0 1 Using that |B| N 2/p, δ = N 2/p−1 and that the integrals are ﬁnite, we arrive at the recursion

p p/2 EωKp (ω) ≤ C1EωKp (ω) + C2 p which implies that EωKp (ω) is bounded by a constant independent of N.

References

[1] Bourgain, J. Bounded orthogonal systems and the Λ(p)-set problem. Acta Math. 162 (1989), no. 3–4, 227–245.

Mark Lewko, UT-Austin / UCLA email: [email protected]

61 10 Local estimates of exponential polynomials and their applications to inequalities of uncertainty principle type - part I

after F. L. Nazarov [3] A summary written by Christoph Marx Abstract A classical result by PálTurán,estimates the global behavior of an exponential polynomial on an interval by its supremum on any arbitrary subinterval. We discuss F. L. Nazarov’s extension of this “global to local reduction” to arbitrary Borel sets of positive Lebesgue measure. More recently, an observation by O. Friedland and Y. Yomdin, enlarges the class of sets to encompass even discrete, in particular finite, sets of sufficient density.

10.1 Introduction We consider an exponential polynomial, i.e. an expression of the form n X λkt p(t) = cke , (1) k=1 where both the coefficients ck and the frequencies λk are complex. The number of non-vanishing coefficients defines its order. Following, depending on the context, µ denotes the Haar measure on R or T := R/Z such that, respectively, µ(T) = 1 or µ([0, 1]) = 1. A classical Lemma due to PálTurán[4] estimates the global behavior of (1) (order n) on an interval I ⊆ R by its supremum on any arbitrary sub-interval E ⊆ I: Aµ(I)n−1 sup |p(t)| ≤ eµ(I)·max |Reλk| · sup |p(t)| . (2) t∈I µ(E) t∈E Here, A > 0 is an absolute constant, independent of n. In particular, comparing (2) with an analogous result for algebraic polynomials dating back to Chebyshev, implies that exponential polynomials of order n behave like their algebraic counterparts of degree n − 1. Following, we discuss the extension of Turán’sLemma to arbitrary Borel sets E ⊆ I, achieved by F. L. Nazarov in [3], Chapter 1 therein:

62 Theorem 1 ([3]). Let p(t) be an exponential polynomial of order n of the form given in (1). Tur´an’s Lemma (2) holds for any Borel set E ⊆ I with µ(E) > 0.

Thus, the above mentioned analogy between exponential and algebraic polynomials persists when considering arbitrary Borel sets, Theorem 1 then being paralleled by Remez type estimates. For an extensive review of available results for both algebraic and exponential polynomials, we refer to e.g. [2]. More recently, Theorem 1 was extended further to certain discrete, in particular ﬁnite, sets of suﬃcient density [1]. To this end, Friedland and Yomdin introduce the metric span of a set E ⊆ I for a given pair (P,I):

Deﬁnition 2 ([1]). Let p(t) be an exponential polynomial of order n of n(n+1) the form given in (1) and I an interval. Set m := 2 + 1, C(n) := 2m 2m2 m(2m + 1) 2 and λ := max |Imλk|. Letting d := C(n)µ(I)λ, introduce d the “frequency bound” M(p, I) := b 2 c + 1. The metric span of a set E ⊆ R is deﬁned by ω(p,I)(E) := sup [M(, E) − M(p, I)] , (3) >0 where M(, E) is the -covering number of E 1.

Remark 3. (i) Clearly, for any measurable E, ω(p,I)(E) ≥ µ(E).

(ii) ω(p,I)(E) > 0 if [M(, E) − M(p, I)] > 0, for some > 0. In particular, ω(p,I)(E) > 0, for discrete sets of suﬃcient density. (iii) The number M(P,I) characterizes the complexity of sub-level sets, {t ∈ I : |p(t)| ≤ δ} (see also Lemma 8, below).

Theorem 4 ([1]). Replacing µ(E) by ω(p,I)(E), the statement of Theorem 1 holds for any E ⊆ R with ω(p,I)(E) > 0. We mention, that Theorem 4 was preceded by an analogous result for algebraic polynomials [5].

1We recall, that given a metric space (M, d) and > 0, one deﬁnes the -covering number of a subset X ⊆ M as the minimal number of -balls needed to cover X.

63 10.2 Nazarov’s theorem Following, we present the main ideas in the proof of Theorem 1. We will focus Pn iλkt on the case when p(t) has purely imaginary frequencies, p(t) = k=1 cke , 1 1 λ1 < ··· < λn. Without loss of generality, we take I = [− 2 , 2 ]. Theorem 1 is based on two crucial Lemmas, one quantifying the number of zeros in a vertical strip (see Lemma 7, below), the other allowing reduction of order by a weak-type estimate of the logarithmic derivative (see Lemma 6, below).

10.2.1 Bernstein-type estimates and order reduction The strategy of order reduction is most transparent when p(t) is a trigono- 2πit metric polynomial, i.e. λk = mk ∈ Z. Substituting z = e , consider Pn mk p(z) = k=1 ckz as a Laurent polynomial on the unit circle. We shall show:

Theorem 5 (see Theorem 1.4 in [3]). Given E ⊂ T, µ(E) > 0, one has

n n−1 X 16e 1 kpkW := |ck| ≤ sup |p(z)| . (4) π µ(E) z∈E k=1 To prove Theorem 5, one inductively reduces the order of p(z) by constructing a sequence of Laurent polynomials p = pn, pn−1, . . . , p1 satisfying

(Ind1) ordpk = k

π (Ind2) kpkkW ≥ 16 kpk−1kW such that pk−1(z) 1 µ z ∈ T : > t ≤ , (5) pk(z) t for 2 ≤ k ≤ n. Pk rs pk−1 is obtained from pk =: s=1 dsz , r1 < . . . rk ∈ Z, choosing one of the following Laurent polynomials of order k − 1 d d q(z) := (z−r1 p (z)) or q(z) := (z−rk p (z)) , (6) dz k dz k

which guarantees a lower bound of k.kW as indicated in (Ind2).

64 Thus, one can reduce the order of p from n to 1,

n−1 π p1(z0) kpkW ≤ kp1k = |p1(z0)| = · |p(z0)| , (7) 16 p(z0) arriving at (4), provided there exists some z0 ∈ E satisfying

n−1 p1(z0) e ≤ . (8) p(z0) µ(E)

Existence of such z0 follows from a measure estimate of the exceptional set where (8) is violated. Noticing that | pk−1(z) | can be realized as a logarithmic pk(z) derivative of an algebraic polynomial, such estimate is accomplished by the following Bernstein-type Lemma:

Lemma 6 (see Lemma 1.2 in [3]). Let g(z) be an algebraic polynomial of degree n. Then, 0 g (z) 8n µ z ∈ T : > y ≤ . (9) g(z) πy

10.2.2 The role of “zero counting” Quantifying the distribution of zeros of exponential polynomials is a crucial ingredient for both the Theorems 1 and 4. Nazarov’s argument is based on the Langer lemma:

Pn iλkz Lemma 7 (see Lemma 1.3 in [3]). Let p(z) = k=1 cke , 0 = λ1 < λ2 < ··· < λn =: λ, be an exponential polynomial not vanishing identically. Then, the number of complex zeros of p(z) in an open vertical strip x0 < Rez < λ∆ x0 + ∆ of width ∆ > 0 does not exceed (n − 1) + 2π .

In particular, based on Lemma 7, one concludes that complex zeros {zj} of the given exponential polynomial p(z) are suﬃciently separated: Ordering zj according to increasing |Rezj|, the inequality

j − (n − 1) |Rez | ≥ π , (10) j (n − 1) holds.

65 We employ the Hadamard factorization theorem,

κ Y Y z p(z) = ceaz (z − z ) 1 − ez/zj =: ceazQ(z)R(z) , (11) j z j=1 j>κ j with κ chosen such that Q(z) contains all zeros of p(z) on [−1/2, 1/2]. By (10), κ can be estimated depending on the relation between λ and n − 1, ( 2(n − 1) , if λ ≤ n − 1 , κ = (12) 2λ , if λ > n − 1 . Using order reduction similar to Sec. 10.2.1, it suﬃces to consider λ ≤ n − 1. The argument principle allows to quantify the contribution of the zero- free part of p(z),

max |ceazR(z)| ≤ 3n−1 min |ceazR(z)| . (13) z∈[−1/2,1/2] z∈[−1/2,1/2]

Finally, the polynomial Q(z) is dealt with using a Cartan-like estimate: Given 0 < h < 1/8, it is shown that for z outside an exceptional subset Ωh ⊆ [−1/2, 1/2] with µ(Ωh) ≤ 8h < 1 = µ(I), one has

|Q(z)| 8h n−1 ≥ √ . (14) max{|Q(t)| : t ∈ [−1/2, 1/2]} 32 3 4 Thus, letting h = µ(E)/8, we may combine all the pieces to arrive at

sup |p(t)| ≤ 3n−1 inf |ceazR(z)| · sup |Q(z)| (15) z∈I z∈I z∈I ( √ )n−1 96 3 4 ≤ inf |ceazR(z)| · inf |Q(z)| (16) µ(E) z∈I z∈E∩Ω ( √ )n−1 96 3 4 ≤ sup |p(z)| . (17) µ(E) z∈E

10.3 Extensions

We conclude by brieﬂy commenting on Theorem 4. For δ := supt∈E |p(t)|, consider the sublevel set Vδ := {t ∈ I : |p(t)| ≤ δ}. Clearly, one has E ⊆

66 Vδ. The main idea in [1] is to reduce Theorem 4 to Theorem 1 by showing ω(p,I)(E) ≤ µ(Vδ) (see Lemma 2.3 in [1]). Again, “counting zeros” provides the key ingredient and also explains the definition of M(p,I): Lemma 8 (see Lemma 2.2 in [1]). For p(t) as in Theorem 4, given η > 0 the number of non-degenerate solutions of the equation |p(t)|2 = η in the interval I does not exceed d = C(n)µ(I)λ. Here, C(n) and λ are defined as in Definition 2.

Lemma 8 allows to estimate from above the -covering number of Vδ, which in turn yields ω(p,I)(E) ≤ µ(Vδ), as claimed.

References

[1] O. Friedland, Y. Yomodon, An observation on the Tur´an-Nazarov inequality., preprint (2012); available on arXiv:1107.0039v2 [math.FA].

[2] M. I. Ganzburg Polynomial inequalities on measurable sets and their applications., Constr. Approx. 17 (2001), 275 - 306.

[3] F. L. Nazarov, Local estimates of exponential polynomials and their applications to inequalities of uncertainty principle type., Algebra i Analiz 5 5 (1993), no. 4, 3 - 66 (Russian); translation in St. Pe- tersburg Math. J. 5 (1994), no. 4, 663 - 717; also available under http://www.math.msu.edu/˜fedja/pubpap.html.

[4] P. Tur´an, Eine neue Methode in der Analyses und deren Anwendungen, Acad. Kiado, Budapest (1953).

[5] Y. Yomdin, Remez-type inequality for discrete sets, Israel Journal of Mathematics 186 (2011), no. 1, 45 - 60.

Christoph Marx, UCI / Caltech email: [email protected]

67 11 Salem sets and restriction properties of Fourier transforms

after G. Mockenhaupt [2] A summary written by Eyvindur Ari Palsson

Abstract We give an analogue on the real line of the restriction phenomena of the Fourier transform which was ﬁrst discovered in the late sixties by E.M. Stein for higher dimensions.

11.1 Classical restriction Restriction problems involve restricting the Fourier transform of a function to a subset of Rn and showing that the restriction is well deﬁned and can be controlled by some norm of the original function. A classical result of this ﬂavor is the Stein-Tomas theorem.

p n 2n+2 Theorem 1. If f ∈ L (R ) where 1 ≤ p ≤ n+3 and n > 1 then

kfbkL2(dµ) ≤ CkfkLp(Rn) where µ is the uniform measure on the unit sphere Sn−1 in Rn. There have been many generalizations of this result, mainly for situa- tions where the unit sphere is replaced by some smooth submanifold of Rn satisfying suitable curvature conditions. Prior to Mockenhaupt’s result [2] one could have believed that restriction was genuinely a higher dimensional phenomena because all the generalizations of the Stein-Tomas result had in common that their setting was in higher dimensions.

11.2 Two notions of dimension In order to do restriction on the real line then we need to find suitable subsets that we can restrict to. With that goal in mind we first discuss two notions of dimension. Fix α > 0 and let E ⊂ Rn. For > 0 define ∞ ! X α Hα(E) = inf rj j=1

68 where the infimum is taken over all countable coverings of E by discs D(xj, rj) with rj < . Then let Hα(E) = lim Hα(E). One can show that there is a →0 unique number a0, called the Hausdorff dimension of E or dimH(E), such that Hα(E) = ∞ if α < α0 and Hα(E) = 0 if α > α0. For integer dimensions then the Hausdorff dimension coincides with the regular notion of dimension, however it allows us to extend the notion of dimension to non-integers. As an example then one can for example show that the Cantor middle third set has Hausdorff dimension log(2)/ log(3). A theorem of Frostman shows that if E ⊂ Rn is a compact set with dimH(E) = α then there is a probability measure µ supported on E satisfy- α ing µ(Br(x)) ≤ Cr , where Br(x) denotes a ball of radius r centered at x. Therefore the β-energy of µ defined as ZZ dµ(y)dµ(x) I (µ) = β |x − y|β is finite as long as β < α. Frostman’s theorem also shows that if Iα(µ) < ∞ for some probability measure µ supported on a compact set E, then dimH(E) ≥ α. Using the fact that on the Fourier side we can write

Z |dµc(ξ)|2 Iα(µ) = c n−α dy Rn |ξ| we conclude that Iα(µ) < ∞ provides some information on the size of dµc. We deﬁne the Fourier dimension of a compact set E ⊂ Rn, denoted by dimF (E), as the supremum of β ≥ 0 such that for some probability measure dµ supported on E |dµc(x)| ≤ C|x|−β/2. As an example then the unit sphere in Rn has Fourier dimension n − 1. Observe that the condition implies that Iα(µ) < ∞ for α < β so we always have dimF (E) ≤ dimH(E).

11.3 Main result

Theorem 2. Let µ be a compactly supported positive measure on Rn which satisﬁes the following properties.

(i) There is β > 0 such that |duc(x)| ≤ C|x|−β/2.

69 α (ii) There is α > 0 such that µ(Br(x)) ≤ Cr for every ball Br(x) of radius r centered at x.

2(2n−2α+β) Then for 1 ≤ p < 4(n−α)+β , we have

kfbkL2(dµ) ≤ CkfkLp(Rn).

Observe that if µ is the uniform measure on the unit sphere Sn−1 in Rn then dimH(E) = dimF (E) = n − 1 and the range of p’s becomes 2n + 2 1 ≤ p < n + 3 which up to the right endpoint recovers the Stein-Tomas theorem.

Note that trivially kfbkL2(dµ) ≤ CkfkL1(Rn) so we are mainly interested in cases where we can have non-trivial bounds. Observe that 2(2n − 2α + β) β = 1 + 4(n − α) + β 4(n − α) + β so we always get some non-trivial bounds. Further note that for ﬁxed α and n then this function is increasing in β so it is of interest to push β ≤ dimF (E) ≤ dimH(E) as high as possible, where E = supp µ, in order to get a maximal range of non-trivial results.

11.4 Salem sets

n Compact sets E ⊂ R that fulﬁll dimF (E) = dimH(E) are called Salem sets, named after R. Salem. We have already seen one such example which is the uniform measure on the unit sphere Sn−1 in Rn. The existence of such subsets on the real line was ﬁrst shown by R. Salem [3]. Another example of a Salem set, due to R. Kaufman [1], is the set Et, t > 0, of those real numbers x ∈ [0, 1] such that

kqxk ≤ q−1−t has solutions for arbitrarily large integers q. Here kxk denotes the distance 2 to the nearest integer. R. Kaufman showed that dimF (E) = dimH(E) = 2+t . J. P. Kahane has further provided a rich class of Salem sets by showing that images of compact sets of a given Hausdorﬀ dimension under Brownian motion are almost surely Salem sets.

70 Going through R. Salem’s construction, which was based on a generalization of the classical Cantor type construction, then one obtains both a set E with dimH(E) = α and a measure dF that fulﬁlls

− α + |dFc (x)| ≤ C|x| 2 .

Now for this particular set and measure we can state a real line analog of the Stein-Tomas result.

2(2−α) Corollary 3. Let 1 ≤ p < 4−3α and dF be as in Salem’s construction for a Salem set on the real line. Then there is a constant C such that

kfbkL2(dF ) ≤ CkfkLp(Rn)

11.5 Sketch of proof of theorem First observe kfbkL2(dµ) ≤ kf ∗ dµckp0 kfkp 1 1 where p + p0 = 1. It is thus suﬃcient to show that the convolution operator 0 T (f) := dµc ∗ f is bounded from Lp → Lp for p0 > 2(2n − 2α + β)/β. ∞ P Let (φk)k=0 be a Littlewood-Paley decomposition, that is, k≥0 φk = 1 and k−1 k P supp φk ⊂ {2 ≤ |x| ≤ 2 }. Then decompose T (f) = k≥0 Tk(f) where

Tk(f) = (φkdµc) ∗ f.

By condition (i) in the theorem then

−k β kTkkL1→L∞ ≤ kφkdµck∞ ≤ C2 2 .

Using Plancherel’s theorem and condition (ii) from the theorem one can show

k(n−α) kTkkL2→L2 ≤ C2 .

Interpolating the two bounds above one obtains

k( 2n−2α+β − β ) p0 2 kTkkLp→Lp0 ≤ C2 .

P 0 Thus T = Tk is bounded for p > 2(2n − 2α + β)/β.

71 References

[1] Kaufman, R., On the theorem of Jarnik and Besicovitch. Acta Arith- metica 39 (1981), 265–267;

[2] Mockenhaupt, G., Salem sets and restriction properties of Fourier transforms. GAFA 10 (2000), 1579–1587;

[3] Salem, R., On singular monotonic functions whose spectrum has a given Hausdorﬀ dimension. Ark. Math. 1 (1950), 353–365;

[4] Wolﬀ, T., Lectures on harmonic analysis With a foreword by Charles Feﬀerman and preface by Izabella Laba. Edited by Laba and Carol Shubin. University Lecture Series, 29. American Mathematical Society, Providence, RI, (2003).

Eyvindur Ari Palsson, University of Rochester email: [email protected]

72 12 Maximal operators and diﬀerentiation theorems for sparse sets: Part II

after I. Laba and M. Pramanik [2] A summary written by Alex Rice

Abstract We state the main results on maximal operators for sets yielded by a random Cantor-type construction. Further, we discuss the deduction of these results from the transverse correlation condition in the main result of Part I.

12.1 Introduction Operators defined by maximal averages have been of central concern in harmonic analysis for many years. Most classically, the Lebesgue differentiation theorem follows from estimates on the Hardy-Littlewood maximal operator, M, defined by 1 Z Mf(x) = sup |f(y)|dy, r>0 |Br(x)| Br(x) where Br(x) denotes the ball of radius r centered at x and | · | denotes Lebesgue measure. More generally, one can consider maximal averages, and deduce differentiation theorems, over sets besides balls. The following result considers the case of spheres, and the proof in dimension d = 2 inspired much of our subsequent discussion.

Theorem 1 (Stein [3]: d > 2, Bourgain [1]: d = 2). Consider the spherical maximal operator on Rd deﬁned by Z ˜ MSd−1 f(x) = sup |f(x + ry)|dσ(y), r>0 Sd−1 where σ is the normalized Lebesgue measure on the unit sphere Sd−1. If ˜ p d d d ≥ 2, then MSd−1 is bounded on L (R ) for p > d−1 , and this range of p is optimal.

Many results of this type are known for other sets under varying smooth- ness and curvature conditions, but no similar theory has developed so far in

73 one dimension. When making connections between harmonic analysis and additive combinatorics, we often make an analogy between curvature and “randomness”, for example with regard to decay estimates on Fourier transforms. With this analogy in mind, we proceed in one dimension with the role of a “curved surface” played by the randomized Cantor-type construction from Part I.

12.2 Main results

For the remainder of our discussion, we let {Sk}k∈N be a decreasing sequence of subsets of [1, 2] yielded by the random Cantor-type construction from Part ∞ I, so the Hausdorff dimension of S = ∩k=1Sk is 1 − with 0 ≤ < 1/3. In particular, we have that |Sk| & 0 as k → ∞, and the densities φk = 1Sk /|Sk| converge weakly to a probability measure µ supported on S. We also assume that {Sk} satisfies the transverse correlation condition in the conclusion of the main result of Part I, which holds with positive probability. Theorem 2. The restricted maximal operators M and M, defined by 1 Z Mf(x) = sup |f(x + ry)|dy 10, k≥1 |Sk| Sk and Z M˜ af(x) = sup ra |f(x + ry)|dµ(y), r>0 R are bounded from Lp(R) to Lq(R). In particular, M˜ := M˜ 0 and M˜ := M˜ 0 p 1+ are bounded on L (R) for all p > 1− .

74 12.3 Motivation In his proof of Theorem 1 in dimension d = 2, Bourgain relies on observations concerning the size of intersections of thin annuli. Roughly, he observes that large intersections between two aﬃne copies of a thin annulus, such as those resulting from “internally tangent” annuli in a clamshell conﬁguration, are extremely rare, and the remaining generic, or transverse, intersections are very small. However, one can see that when considering “square annuli”, large intersections are much more common due to the lack of curvature. One might expect that an adaptation of Bourgain’s method is possible as long as some analog of these observations on intersections holds. Motivated by the analogy between curvature and randomness, we turn to the transverse correlation condition, analogous to Gowers’ higher order uniformity from additive combinatorics, to adapt Bourgain’s method to the random Cantor- type construction.

12.4 Linearization and discretization

Recall from the construction that each set Sk is a union of Pk intervals of length δk. While we don’t discuss the details here, one can establish the maximal estimates in Theorem 2 by examining the linearized and discretized auxiliary operators Φk deﬁned by Z Φkf(x) = f(z)Vk,x(z)dz, R

z−c(x) where Vk,x(z) = σk r(x) , σk = φk − φk+1, and c(x) and r(x) are functions taking values in discrete, equally spaced sets C ⊆ [−4, 0] and R ⊆ [1, 2], respectively. Further, Φkf is supported in [−4, 0] and c(x) satisﬁes

−1 |x − c(x)| ≤ δk ≤ Pk for all x ∈ [−4, 0]. (2)

p q L [0, 1] → L [−4, 0] operator norm bounds on Φk can be deduced from estimates on ∗ kΦk1Ωkn sup n−1 , (3) Ω⊆[−4,0] |Ω| n ∗ where Φk is the “adjoint” operator deﬁned by Z ∗ Φkg(z) = g(x)Vk,x(z)dx. R 75 As discussed in Part I, one must allow n in (3) to be arbitrarily large in order to obtain the full results of Theorem 2, but here we restrict to n = 2 for a simpliﬁed exposition.

12.5 Transverse correlations

Recall that for a fixed k and a pair of affine transformations g1, g2, we consider the collection of internal tangencies, Fint, which roughly corresponds to pairs of intervals I1,I2 which are close together and satisfy g1(I1) ∩ g2(I2) 6= ∅. Of course, when dealing with affine transformations we can equivalently consider translation-dilation pairs, where (c, r) ∈ R2 corresponds to g(x) = rx + c, and here we consider the collection of pairs of transformations n o Uk = (c1, r1), (c2, r2) : c1, c2 ∈ C, r1, r2 ∈ R , where C and R are as in section 12.4. We partition this collection into the pairs of transformations which have many internal tangencies, and those which do not. Specifically, we let

int p Uk = {A ∈ Uk : #(Fint) ≥ Pk} and tr int Uk = Uk \Uk .

Recall that for A ∈ Uk and functions f1, f2 on R, we deﬁne the correlation of f1, f2 according to A by Z z − c1 z − c2 Λ(A; f1, f2) = f1 f2 dz, R r1 r2 and we write Λ(A; f) for Λ(A; f, f). The main component we wish to illumi- nate is the following, which states that control on correlations for transverse ∗ pairs according to σk yields the desired type of estimate on Φk. Proposition 3. If sup |Λ(A; σk)| ≤ C0(k), (4) tr A∈Uk then ∗ −1/2 1/2 kΦk1Ωk2 n Pk o sup 1/2 ≤ C max ,C0(k) Ω⊆[−4,0] |Ω| |Sk+1| for an absolute constant C > 0.

76 The main result of Part I can be used to show that (4) holds with C0(k) decaying exponentially in k, which is suﬃcient for the purposes of Theorem int 2. When encountering pairs in Uk , we use the following two facts, the ﬁrst of which allows us to conclude that such pairs are rare, and the second of which provides a trivial upper bound on the relevant correlation.

Lemma 4. If A = {(c1, r1), (c2, r2)} ∈ Uk and #(Fint) ≥ L, then

|c1 − c2| ≤ min{4, 160/L}. While we omit the details of Lemma 4, we do provide a proof of the following bound which ignores any potential cancellation in the integral deﬁning Λ(A; σk).

Lemma 5. For all k ≥ 1 and A ∈ Uk, we have 8 |Λ(A; σk)| ≤ . |Sk+1|

Proof. Recalling that σk = φk+1 −φk, we see that Λ(A; σk) expands as a sum of four terms of the form ±Λ(A; φk+λ1 , φk+λ2 ) with λ1, λ2 ∈ {0, 1}, and we estimate each in absolute value. Using that the sequence |Sk| is decreasing and the scaling parameters ri ≤ 2, we have 1 Z z − c z − c |Λ(A; φ , φ )| = 1 1 1 2 dz k+λ1 k+λ2 Sk+λ1 Sk+λ2 |Sk+λ1 ||Sk+λ2 | R r1 r2 1 Z z − c 2 ≤ 1 2 dz ≤ , Sk+λ2 |Sk+1||Sk+λ2 | R r2 |Sk+1| and the lemma follows. Proof of Proposition 3. Fixing Ω ⊆ [−4, 0], we see Z 2 ∗ 2 kΦk1Ωk2 = Vk,x(·)dx Ω 2 Z Z Z = Vk,x1 (z)dx1 Vk,x2 (z)dx2 dz R Ω Ω Z Z = Vk,x1 (z)Vk,x2 (z)dz dx1dx2 Ω2 Z R = Λ(A(x1, x2); σk)dx1dx2 Ω2 Z = Λ(A(x1, x2); σk)dx1dx2, Θ1tΘ2

77 where A(x1, x2) = (c(x1), r(x1)), (c(x2), r(x2)) ∈ Uk,

2 int Θ1 = {(x1, x2) ∈ Ω : A(x1, x2) ∈ Uk }, and 2 tr Θ2 = {(x1, x2) ∈ Ω : A(x1, x2) ∈ Uk }.

First we estimate the integral over Θ1, on which we can only trivially bound the integrand, but we see that the integration is restricted to a small set. Speciﬁcally, we have by Lemma 4 that

n 2 −1/2o Θ1 ⊆ (x1, x2) ∈ Ω : |c(x1) − c(x2)| ≤ 160Pk

n 2 −1/2o ⊆ (x1, x2) ∈ Ω : |x1 − x2| ≤ 320Pk , where the last inclusion uses (2). In particular we know that −1/2 |Θ1| ≤ 640Pk |Ω|, which combined with Lemma 5 yields Z −1/2 5120Pk |Λ(A(x1, x2); σk)|dx1dx2 ≤ |Ω|. Θ1 |Sk+1| Further, the correlation condition (4) immediately yields Z 2 |Λ(A(x1, x2); σk)|dx1dx2 ≤ C0(k)|Ω| ≤ C0(k)|Ω|, Θ2 and the proposition follows.

References

[1] Bourgain, J. Averages in the plane over convex curves and maximal operators. J. Analyse Math. 47 (1986), 69-85. [2] Laba, I. and Pramanik, M. Maximal operators and diﬀerentiation theorems for sparse sets. Duke Math. J. 158 (2011), 347-411. [3] Stein, E. M. Maximal functions: Spherical means. Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 2174-2175. Alex Rice, UGA email: [email protected]

78 13 Maximal operators and diﬀerentiation theorems for sparse sets, part I

after I.Laba and M. Pramanik [1] A summary written by Pablo Shmerkin

Abstract The proof of Lp bounds for maximal operators associated to a sequence {Sk} of sparse sets depends critically on a multiscale higher- order uniformity condition, given in terms of a bound on certain correlation integrals. I sketch the proof of these correlation bounds for a sequence {Sk} constructed via a random Cantor iteration. In con- junction with the second part, this establishes the existence of sparse sets for which the corresponding maximal operators are bounded from Lp to Lq for appropriate choices of (p, q).

13.1 Introduction

Let {Sk : k ≥ 1} be a decreasing sequence of compact subsets of R. Associ- ated to this sequence is the maximal operator 1 Z Mff (x) = sup |f(x + ry)|dy. r>0,k≥1 |Sk| Sk

p (|Sk| is the Lebesgue measure of Sk.) In order to obtain L bounds for this and related operators, a critical step is to obtain good estimates for correlations of the form Z n Y −1 Λ(g1, . . . , gn; σk) = σk(g` (z))dz, (1) `=1 where 1Sk+1 1Sk σk = − |Sk+1| |Sk| measures the “oscillation” of the sequence {Sk}, and g1, . . . , gn are affine functions. The idea is that for “transversal” tuples of affine functions there should be a large amount of cancelation in the integral, provided the sets Sk are sufficiently “random”. At the other extreme, if the g` are “tangential” (i.e. some of them are nearly equal) then one cannot do better than the trivial

79 estimate for the integral, but on the positive side there will be “relatively few” such bad tuples. In this note we define a random construction of Cantor iteration type, and prove efficient bounds for the correlations Λ(g1, . . . , gn; σk) under appropriate transversality conditions on the affine maps.

13.2 Construction of the sets Sk

We start by describing the construction of the sets Sk. The specifics of the random choices differ slightly from those in [1], and help avoid a number of technical steps in the proofs and reduce the number of parameters involved. The construction will depend on an integer N ≥ 2 and a sequence {k} of numbers in (0, 1). For simplicity, we also assume that N 1−k ∈ N for all k. Informally, the sets Sk will be constructed as follows: we subdivide [1, 2] 1−1 into N equal subintervals, and pick N of them at random; let S1 be the union of these intervals. We next subdivide each of the intervals that 1−2 make up S1 into N subintervals of equal length, and pick N of them at random, with all the choices independent. We let S2 be the union of the picked intervals of this level, and so on. We now give a formal definition that will also allow us to define useful notation. Let

I = Ik = {i = (i1, i2, . . . , ik) : 1 ≤ ir ≤ N, 1 ≤ r ≤ k}.

Elements of Ik will index N-adic intervals of step k in the construction; the left point of the interval associated to i ∈ Ik is i − 1 i − 1 α(i) = 1 + 1 + ... + k , N N k

−k and the length of the interval is δk = N . The interval in question is then Ik(i) = [α(i), α(i) + δk]. Next we deﬁne, for each k, a collection of indicator random variables {Xi : i ∈ Ik} taking values in {0, 1} that will describe the collection of “chosen” intervals. To begin, let X1 be a random subset of {1,...,N} with 1−1 N elements (the choice is uniform over all such subsets), and let X1 be the indicator function of X1. Now suppose the family {Xk(i): i ∈ Ik} has been deﬁned. Let {Yi : 1−k+1 i ∈ Ik} be independent random subsets of {1,...,N} with N elements, write Yi for the indicator function of Yi, and set Xk+1(ia) = Xk(i)Yi(a).

80 (Here and throughout, ia denotes the juxtaposition (i1, . . . , ik, a).) Finally, S we set Sk = Ik(i). Let i∈Ik:Xi=1 k Y 1−i Pk = N . i=1

Then Sk is made up of Pk intervals of length δk. The following facts are either immediate from the deﬁnitions, or follow from standard arguments: T∞ 1. The sequence Sk is decreasing; let S = k=1 Sk.

2. The densities 1Sk /|Sk| converge weakly to a measure µ supported on S. 1 Pk 3. The Hausdorﬀ dimension of S is dimH S = lim infk→∞ k i=1(1 − i).

13.3 Internal tangencies and transversal intersections In order to obtain the most general results, it is important to allow n in (1) to be arbitrarily large. However, the case n = 2 is enough to obtain L2+δ estimates and already requires the main ideas needed to tackle the general case, while being notationally much simpler. We therefore restrict ourselves to n = 2 in the sequel. The pair of affine functions in (1) will ultimately be drawn from certain discrete family, but most of the analysis will be for a fixed pair g1, g2. We always assume that gi(x) = rix + ci, where ri ∈ [1, 2] and ci ∈ [−4, 0]. We index the pairs of intervals which have nonempty intersection after being mapped by g1, g2: 2 F = (i1, i2) ∈ Ik : g1(Ik(i1)) ∩ g2(Ik(i2)) 6= ∅ . It will be critical to distinguish between two kinds of intersections: those in which the intervals Ik(i1),Ik(i2) are close to each other (at a distance bounded by a constant multiple of their size), and the rest. The precise definition is as follows. Let

Fint = {(i, j) ∈ F : ii = ji for i ≤ k − 1, |ik − jk| ≤ 4} . This collection is referred to as the class of internal tangencies. Its complement Ftr = F \ Fint will be called the class of transversal intersections. It can be checked that, in a precise way, a large number of internal tangencies forces the maps g1 and g2 to be close, and viceversa. We say that the pair (g1, g2) is transversal if the number√ of internal tangencies is small in the following quantitative sense: #Fint < Pk.

81 13.4 Main result We can now state the main result.

tr Theorem 1. For each k, let Uk be a family of transversal pairs (g1, g2) with −2L at most A := 4δk+1 elements for some constant L ≥ 1 (these families and the constant L will be deﬁned in the second part). There exists C > 0 such that with positive probability (on the choices of X1, Yi), the following holds for all k:

Pk p 2k+1 − (1+3j )/2 sup Λ(g1, g2; σk) ≤ C Lk log NN N j=1 . (2) tr (g1,g2)∈Uk We make some remarks:

1. Given ∈ [0, 1/3), with appropriate choices of the parameters we can ensure that dimH S = 1 − , |S| = 0, and the right-hand side of (2) decays exponentially in k.

2. There is a corresponding version for transversal n-tuples (g1, . . . , gn) that we omit, but is needed to prove the most general Lp → Lq estimates.

3. The bounds obtained can be seen as a multiscale analog to second order uniformity conditions in additive combinatorics (for larger values of n one would obtain higher order uniformity-type estimates).

In the rest of the note we outline the proof of Theorem 1, trying to emphasize the main ideas and skipping all calculations. The proof goes by conditioning on Sk (or, more precisely, on {Xi : i ∈ Ik}), and proving that (2) holds with large probability for k + 1. A little algebra shows that

1 N X X −k+1 σk(z) = Xk(i) (Yi(a) − N )1Ik+1(ia)(z). Pk+1δk+1 i∈Ik a=1 From here it is easy to deduce that

1 X Λ(g1, g2; σk) = 2 Γ(i, j), (3) (Pk+1δk+1) (i,j)∈F:Xk(i)Xk(j)=1

82 N X −k+1 −k+1 Γ(i, j) = (Yi(a) − N )(Yj(b) − N )|g1(I(ia)) ∩ g2(I(jb))|. a,b=1

Because we are conditioning on Sk, the randomness comes in the Γ(i, j). We int tr split the sum (3) as Ξk +Ξk , where the ﬁrst corresponds to pairs (i, j) ∈ Fint and the second in Ftr. Under the assumption that (g1, g2) is transversal (i.e. 1/2 #Fint < Pk ), simple geometric arguments yield a deterministic bound

1/2 int 4δkPk Ξk ≤ 2 . (Pk+1δk+1)

13.5 A martingale argument, and conclusion of the proof

tr −k+1 The crux of the proof is to estimate Ξk . Since Yi(a) has mean N for all i ∈ Ik and a ∈ {1,...,N}, the random variables Γ(i, j), (i, j) ∈ Ftr all have zero mean (recall that i 6= j for (i, j) ∈ Ftr). If they were also independent, the desired estimates would follow from standard exponential concentration bounds, but this is not the case: Γ(i1, j1) and Γ(i2, j2) are correlated whenever {i1, j1} ∩ {i2, j2} 6= ∅. The main idea to handle this issue is to decompose (i) Ftr into a bounded number of subsets Ftr , such that each sum

X (i) {Γ(i, j): XiXj = 1, (i, j) ∈ Ftr } can be reordered as a martingale (recall that a sequence of random variables ∞ {Mr}r=0 is a martingale if E(Mr+1|Br) = Mr for all r, where {Br} is an increasing ﬁltration of σ-algebras). Martingales satisfy concentration bounds essentially as good as sums of iid random variables. A classical example is:

Theorem 2 (Azuma’s Inequality). Let {Mr} be a martingale, and suppose |Mr+1 − Mr| ≤ cr a.s. for some positive numbers cr. Then for all T ∈ N and all λ ∈ , R ! λ2 (|MT − M0| ≥ λ) ≤ 2 exp − . P PT 2 2 r=1 cr

(i) The families Ftr are obtained as follows. An easy consequence of the deﬁnition of Ftr is that the projections (i, j) → i, (i, j) → j are at most 4- to-1 on Ftr, so we can decompose Ftr into 16 classes, on each of which the

83 projections are both injective. We further subdivide each of these classes into two subsets, so that on each class αk(i) − αk(j) is either everywhere positive or everywhere negative (by transversality it cannot be 0). (i) Let Ftr be one of the classes constructed in this way, and without loss (i) (i) of generality suppose αk(i) < αk(j) for all (i, j) ∈ Ftr . We enumerate Ftr = {(i1, j1),..., (iT , jT )}, where αk(jt) is strictly increasing in t (there can be no repetitions thanks to the injectivity of the projection). Further, let Br be the σ-algebra generated by {Zi : α(i) < α(jr)}. One can then check that Pr the sequence M0 = 0, Mr = t=1 Γ(it, jt) for 1 ≤ r ≤ T , is a martingale. The essential reason for this is that at most one set {it, jt} (1 ≤ t < r) can intersect {ir, jr}. p 2 Applying Azuma’s inequality with λ = 4δk 2Pk log(200k A) we get, after some easy estimates,

X (i) 1 {Γ(i, j): X X = 1, (i, j) ∈ } > λ ≤ . P i j Ftr 100k2A

So far the analysis has been for a fixed transversal pair (g1, g2). Now, since tr (i) by assumption #Uk ≤ A, and there are at most 32 classes Ftr , we conclude that tr 32λ 1 P Ξk > 2 ≤ 2 . (δk+1Pk+1) 3k P 1 As k 3k2 < 1, the proof is finished after a little more algebra to combine the tr int bounds for Ξk and Ξk into the√ form given in the statement of the theorem (we note that with the bound Pk in the definition of transversal pair (g1, g2), tr int the estimates for Ξk and Ξk are essentially equal).

References

[1]Laba, I. and Pramanik, M., Maximal operators and diﬀerentiation theorems for sparse sets. Duke Math. J. 158 (2011), no. 3, 347–411;

Pablo Shmerkin, University of Surrey email: [email protected]

84 14 Bounded orthogonality systems and the Λ(p)-set problem I

after Jean Bourgain [1] A summary written by Stefan Steinerberger

Abstract We describe a pivotal result of Jean Bourgain in the theory of Λ(p)−sets; it states that the property of being Λ(p) is - in a certain sense - a generic property and highlights the eﬃciency of interfacing elements from analysis and probability theory.

14.1 Introduction

We consider the one-dimensional torus T and the standard Lp−norm. Deﬁnition 1. Let p > 0. A set E ⊂ Z is called a Λ(p) set if there is a 0 < q < p such that for all functions f : T → C with supp fˆ ⊂ E the inequality kfkLp . kfkLq holds for an absolute implicit constant depending only on E. If this property holds for some 0 < q < p, then it is known to holds for all 0 < q < p (with the implicit constant depending also on q) thereby justifying the name Λ(p)-set. The origin of this deﬁnition can be traced back to an old conjecture, which states that the squares {k2 : k ∈ N} are a Λ(p) set for all p < 4, or, alternatively,

N N 2 2 X 2πik x X 2πik x ake .p ake (1) p 2 k=1 L (T) k=1 L (T) Early results are due to Rudin, who gave explicit constructions based on the following theorem.

Theorem 2 (Rudin, 1960). Let 1 < s ∈ N and E ⊂ N. If there is a constant C < ∞ such that any integer n ∈ N has at most C representations as the sum of s elements of E, then E is a Λ(2s) set.

85 Based on results of this type exploiting the combinatorial properties of the Lp norm for p ∈ 2N, it was shown that there exist Λ(2n)-sets which are not Λ(2n+ε)-sets for any ε > 0. We know from results of Bachelis-Ebenstein, Rosenthal and Hare that this is wrong for any Λ(p)−set with 0 < p < 2. The main problem has therefore been the construction of sets showing that for p ≥ 2 the inclusion of Λ(p + ε) ⊆ Λ(p) is strict.

This problem was resolved by Jean Bourgain as a consequence of a much more powerful theorem with many implications on the structure of Λ(p)−sets. His theorem holds in the more general context of uniformly bounded orthogonal systems - a simplified version can be stated as follows. Theorem 3 (Bourgain, 1989, simplified). Let 2 n2/p such that E is a Λ(p)−set. In fact, this property holds for generic sets E of size n2/p. Remarks. 1. It is easy to see that n2/p is the maximal size of a Λ(p) set contained in {1, 2, . . . , n}. 2. Standard results from harmonic analysis allow to extend the result to build infinite Λ(p)-sets which, by density arguments alone, are Λ(p) but not Λ(p + ε). 3. The theorem implies that being a Λ(p)-set is less special a property than previously thought - however, it has no implications for specific sets (i.e. squares). Λ(p)−sets have applications in the study of Lp−improving measures on T whose convolution with a Lp function yields a smoother Lp+ε function - this property is known to be equivalent to ’large’ Fourier coefficients of the measure having a Λ(p)−type structure.

Conversely, arguments in the early style of Rudin have matured into statements linking the multiplicative structure and the Λ(p)−constant. Theorem 4 (Bourgain & Chang, 2003). Given ε > 0 and p > 2, there exists δ(ε, p) > 0 such that for any set A ⊂ Z the growth bound |A2| ≤ |A|1+ε implies δ λp(A) ≤ |A| , where δ → 0 as ε → 0.

86 14.2 Sketch of the general proof Very roughly summarized, the proof can be said to use a decomposition into big and small terms to get a bootstrap argument on the size of the Λ(p)−constant; for the bootstrap to close one needs to estimate complicated expressions. The core element of the proof is to exploit the fact that by randomizing over sets the structure of these expressions can be dealt with in an averaged sense.

We demonstrate the way the bootstrap is set up for 2 < p ≤ 3 and deﬁne for any set S ⊂ {1, 2, . . . , n}

X 2πix KS = sup k aie kLp(T). kak2≤1 i∈S Fix a number 0 < γ < 1 satisfying (1 − γ2)(p−2)/2 + γp < 1 and partition {1, 2, . . . , n} (up to 1 point) into sets I,J such that

min |ai| ≥ max |aj| i∈I j∈J

X 2 2 X 2 2 ai < γ and aj < 1 − γ . i∈I j∈J Having our partition, we deﬁne X X x(u) = aiφi(u) y(u) = ajφj(u) i∈I j∈J and wish to estimate Z |x(u) + y(u)|pdu using (valid for 2 < p ≤ 3) |x + y|p ≤ |x + y|2|y|p−2 + (1 + |x|)p + 2x(1 + |x|)p−2y + (1 + |x|)p−2y2. We arrive at Z p 2 p p−2 p−2 |x(u) + y(u)| du ≤ kx + ykp + k1 + |x|kp + 2 y, x(1 + |x|) + y, y(1 + |x|) !(p−2)/2 !p/2 2 p−2 X 2 p X 2 ≤ KSKS aj + KS ai J I p−1 p−2 p−2 + CKS + 2 y, x(1 + |x|) + y, y(1 + |x|) .

87 Now the partition becomes valuable:

(1 − γ2)(p−2)/2 + γp < 1, which allows a rearrangement of the type

p p−1 KS ≤ CKS + other terms, where the other terms are precisely of the scalar-product-type.

The next steps consist of introducing a dummy variable z(u) (inheriting properties given to x(u), y(u) by virtue of the partition procedure) and using decoupling inequalities. This leads ultimately to terms that can be dyadically decomposed into expression where the combinatorial nature is well embod- ied in the entropy-type structure - the above lemma then reduces things to studying certain geometric properties of the function space when regarded as embedded in a high dimensional Euclidean space (i.e. its diameter and distance relations); these properties can be studied using Bessel’s inequality and certain algebraic inequalities.

References

[1] Bourgain, J. Bounded orthogonal systems and the λ(p)-set problem. Acta Math. 162 (1989), no. 3–4, 227–245.

[2] Bourgain, Jean; Chang, Mei-Chu. On the size of k−fold sum and product sets of integers. J. Amer. Math. Soc. 17 (2004), no. 2, 473–497

[3] Rudin, Walter. Trigonometric series with gaps. J. Math. Mech. 9 (1960), 203–227.

Stefan Steinerbegrer, University Bonn email: [email protected]

88 15 On a problem of Erd˝oson sequences and measurable sets, & Inﬁnite patterns that can be avoided by measure.

after K. Falconer [3] and M. Kolountzakis [4] (respectively) A summary written by Krystal Taylor

Abstract Erd˝osconjectured that given an infinite set A of real numbers, there always exists a measurable set of positive measure which contains no affine copy of A. Some partial progress has been made on solving this problem. In particular, K. Falconer proves this conjecture in the case that A is given by a not-too-rapidly-decreasing sequence of real numbers. M. Kolountzakis shows that for every infinite set A, there is a set E of positive measure such that x + tA ⊂ E fails for almost all (Lebesgue) pairs (x, t).

15.1 Introduction: known classes of non-universal sets Deﬁnition 1. A set A of real numbers is called universal (in measure) if for every measurable E ⊂ R with positive Lebesgue measure there are x, t ∈ R such that x + tA := {x + ta : a ∈ A} ⊂ E.

One can verify that all finite sets of reals are universal. Erd˝os [2] asked whether there exist any infinite universal sets. While no universal sets are known, there are some classes of infinite sets which have been shown to be not universal. Komjáth[5] proves that for every infinite set A ⊂ [0, 1], there is another subset of [0, 1], of measure arbitrarily close to 1 that does not contain any translate of A (no dilations allowed). Kolountzakis [4] gives an alternative proof of Komjáth’sresult. Bourgain [1] explores a 3-dimensional version of the problem for sets with Cartesian product structure. He shows that any given set in R3 of the type S × S × S, where S ⊂ R is infinite, is not universal, defining universality in 3 dimensions. His method allows for non-isotropic scaling along the three axis.

89 Furthermore, he shows that for sets of real numbers of the type S1 + S2 + S3, where the Sj are inﬁnite, are not universal. He points out that a variant of his method shows that sums such as

{2−n} + {2−n} (1) are not universal. Bourgain uses a probabilistic construction. Falconer proves that any sequence {xn} of positive reals decreasing to 0 and satisfying x lim n+1 = 1 (2) n→∞ xn is not universal. He constructs a probabilistic Cantor-type set with positive measure which avoids all affine copies of a given sequence {xn} satisfying (2). His result is stated below in Theorem 2. The problem in the case of geometrically decreasing sequences is open. That is, no non-universal sequence xn ↓ 0 is known which satisfies xn+1 ≤ ρxn, for some fixed ρ < 1. Moreover, it is not even known whether all uncountable sets are not universal [4]. Kolountzakis [4] proves that almost all copies of a given set of real numbers can be avoided by some set of positive measure. His results are stated below in Theorems 3 and 4. As mentioned above, Komjáth’sresult follows as a simple consequence [4].

15.2 Results 15.2.1 Sequences with ’slow decay’ are universal In this section, we present a result of Falconer which classiﬁes all sequences with slow decay as non-universal.

∞ Theorem 2. Let {xn}n=1 be a decreasing sequence of real numbers convergent to 0 such that x lim n+1 = 1. n→∞ xn Then there exists a closed set E with µ(E) > 0 such that for any numbers b, c, with c 6= 0, cxn − b∈ / E for inﬁnitely many n.

90 15.2.2 Infinite sets are ’almost everywhere’ universal Here we present two results due to Kolountzakis which say that almost all copies of an infinite set of real numbers can be avoided by some set of positive measure. Theorem 4 below is a strengthening of Theorem 3, but the proofs are different.

Theorem 3. Let A ⊂ R. There is a set E ⊂ [0, 1] of measure arbitrarily close to 1 such that the set of pairs

{(x, t):(x + tA) ⊂ E} has measure 0 (Lebesgue measure in R2). Theorem 4. Let A ⊂ R. There is a set E ⊂ [0, 1] of measure arbitrarily close to 1 such that

µ{t : ∃ x such that (x + tA) ⊂ E} = 0.

Here and throughout, µ denotes the Lebesgue measure in R.

15.3 Sketch of proofs 15.3.1 Construction of the set E in Theorem 2 The purpose of this section is to present some of the ideas in the proof of Falconer’s Argument. For a given sequence satisfying the conditions of the theorem, Falconer constructs a set E and shows that, for any integer m, there exists an integer n(m) so that

∞ \ 1 (E + b) ⊂ {0} (3) xn n=n(m) for all real numbers b. As a consequence, for any c 6= 0, there exists n ≥ n(m) so that cxn ∈/ E +b. Iterating this process generates inﬁnitely many elements of the sequence for which cxn ∈/ E + b.

To construct E, choose numbers λk (1 ≤ k < ∞) such that 0 < λk < 1 P∞ 1 and k=1 λk < 2 . Let {lk} be a rapidly decreasing sequence of lengths to be determined. Let ∞ Ek = ∪r=−∞ [rlk, rlk + lk(1 − λk)]

91 T∞ and let E = k=1 Ek. Provided that lk < 1, for 1 ≤ k < ∞, it is immediate 1 that µ(E) > 0. To see this, observe that [0, 1]\Ek contains at most ( + 1) lk intervals of length lk, and so

∞ ∞ X X 1 µ([0, 1]\E) ≤ µ([0, 1]\Ek) ≤ λklk( + 1) < 1. lk k=1 k=1

By observing the overlapping of certain intervals, Falconer shows that

∞ \ 1 (Ek + b) ⊂ −2lk/xn(k), 2lk/xn(k) , (4) xn n=n(k) whenever 0 ≤ b ≤ lk. Then by the periodicity of Ek, (4) holds for all real b. One can observe that it is in this aforementioned overlapping that the decay condition is used and that the argument fails for geometric sequences.

Proving (4) reduces to showing that for each k and for each 0 ≤ b ≤ lk,

∞ c lk(1 − λk) + b [ 1 , ∞ ⊂ (Ek + b) , (5) xn(k) xn n=n(k)

c where we recall that 0 < λk < 1. Here and throughout A denotes the com- pliment of the set A in R The idea now is to ﬁx k ∈ N, ﬁx 0 ≤ b ≤ lk, and write the interval on the left-hand-side of (5) as a union of overlapping intervals, each of which lies c in some 1 (E + b) for some n ≥ n(k). It is at this point that the decay xn k condition on the sequence {xn} plays a role in the argument.

15.3.2 Construction of the set E in Theorem 3 The purpose of this section is to present some of the ideas in the proof of Kolountzakis’s Argument.

First, we observe that is is suﬃcient to prove the theorem in the case that A is a sequence of positive reals decreasing to 0. Next, ﬁx an interval [α, β],

92 where 0 < α < β < ∞, for the scaling parameter. The random Cantor-type set, E, is deﬁned as an intersection

∞ \ E = Fj, j=1 where Fj is deﬁned by dividing the unit interval into mj equal subintervals and keeping each of them independently and with equal probability pj. The probabilities pj are taken such that

∞ ∞ Y Y j pj = q and pj = 0, j=1 j=1 something which is possible for every q ∈ [0, 1]. The integers mj are deﬁned to be large enough that ta1, ··· , taj (6) 1 are in separate intervals of Fj for t ∈ [α, β]. In particular, is smaller than mj half of the minimum gap (the largest interval containing no points) of the numbers αa1, ··· , αaj. Now, for a ﬁxed x ∈ [0, 1], it holds that x ∈ E if x is in exactly one of the intervals making up Fj for each value of j. Therefore

∞ Y P r(x ∈ E) = pj = q. j=1 Therefore, Eµ(E) = q.

The next step is to show that

Eµ{(x, t): x + tA ⊂ E} = 0, (7) for any x ∈ [0, 1] and t ∈ [α, β]. ( Here, we are abusing notation by using µ to denote either the Lebesgue measure on R or R2 depending on the context.) It would follow from (7) that

µ{(x, t): x + tA ⊂ E} = 0 (8) almost everywhere, and we could conclude then that there exists a set E, with µ(E) ≥ q which satisﬁes (8), thus concluding the proof of the theorem.

93 Let φ(x, t) be the indicator function of the set of pairs (x, t) so that x + tA ⊂ E. Observe that Z 1 Z β Eµ{(x, t): x + tA ⊂ E} = E φ(x, t)dtdx 0 α Z 1 Z β = Eφ(x, t)dtdx 0 α = 0.

To prove (7), it suﬃces to show then that Eφ(x, t) = 0. For the set x + tA to be contained in E, it is neccessary that all intervals of stage j which contain a point in (6) must be kept. Since there are exactly j such intervals, j the probability of this happening is exactly pj, and the probability of this ∞ Y j happening for all stages is pj = 0. j=1

References

[1] Bourgain, J. Construction of sets of positive measure not containing an aﬃne image of a given inﬁnite structure. Israel J. Math. 60 (1987) 333- 344.

[2] Erd˝os,P. My Scottish book ’problems’. The Scottish Book (ed. R.D. Mauldin, Birkhauser), Boston, 1981, 35-43.

[3] Falconer, K. On a problem of Erd˝oson sequences and measurable sets. Proc. Amer. Math. Soc. 90 (1984), 77-78.

[4] Kolountzakis, M. Inﬁnite patterns that can be avoided by measure. Bull. London Math. Soc. 29 (1997), 415-424.

[5] Komj`ath, P. Large sets not containing images of a given sequence. Canad. Math. Bull. 26 (1983), 41-43.

Krystal Taylor, Technion email: [email protected]

94 16 Averages in the plane over convex curves and maximal operators

after J. Bourgain [2] A summary written by Joshua Zahl

Abstract We prove certain Lp → Lp bounds on the Bourgain circular maximal function.

16.1 Introduction 16.1.1 Background In [1], Bourgain deﬁned the “circular maximal function”: for a bounded measurable function f : R2 → R, the circular maximal function of f is given by 1 Z Mf(x) = sup f(y)dy, (1) t>0 2πt C(x,t) where C(x, t) is the circle centered at x of radius t, and dy is the 1–dimensional arclength measure on the circle C. Bourgain established the bound

kMfkp ≤ Cp kfkp , 2 < p < ∞. (2) This result has several consequences:

1. Diﬀerentiation theorems: Deﬁne 1 Z Atf(x) = f(y)dy. (3) 2πt C(x,t)

Then if f is a bounded measurable function,

f = lim Atf a. e . (4) t→0

2. “circle sets”: Let K ⊂ R2 be a compact set such that for each x ∈ [0, 1]2, K contains a circle centered at x. Then K has positive Lebesgue measure.

95 3. Estimates on the wave equation in 2+1 dimensions: Let u: R+ × 2 R → R be a solution to the wave equation u = 0, u(0, x) = 0, ut(0, x) = f(x). Then we obtain certain estimates on u of the form

ku(t, x)k ∞ p1 2 kfk p2 2 . (5) Lt (R)Lx (R ) . Lx (R ) See i.e. [4], Section 3.2 for more details.

16.1.2 New Results In this paper [2], Bourgain proves an analogue of (2) for a more general class of curves. Let Γ be the boundary of a compact, convex, centrally symmetric set in R2. Assume furthermore that Γ is a smooth curve and has non-vanishing curvature. Let σ be the arclength measure on Γ. For 0 < t < ∞, and for f a bounded measurable function on R2, deﬁne Z AΓ,tf(x) = f(x + ty)σ(dy). (6)

Thus if Γ is the unit circle, then AΓ,tf(x) = Atf(x), where the latter operator is as deﬁned in (3). Deﬁne the maximal operator

MΓf(x) = sup AΓ,tf(x). (7) t>0

The main result of [2] is the following theorem.

Theorem 1. Let f be a bounded measurable function on R2. Then

kMΓfkp ≤ CΓ,p kfkp , 2 < p < ∞. (8)

2 When Γ is the unit circle, then the operator MΓ is not bounded in L . Thus the Lp bounds from Theorem 1 are best possible. Rather than giving a sketch of Theorem 1 in its full form, we will only consider the case where Γ is the unit circle. The proof of this special case captures all of the main ideas of the general proof, but it avoids many of the messy plane geometry details that need to be considered in the full theorem.

96 16.2 Proof the Theorem 1 As noted in the introduction, we will only consider the case where Γ is the unit circle. Thus we will use the operators M and At rather than MΓ and AΓ,t.

16.2.1 Reduction to a geometric problem In this section, we use a variety of (by now standard) techniques from harmonic analysis to reduce the problem of bounding kMfkp to a quantitative form of the following geometric question: Question 2. Given a collection of circles C, such that for each point x ∈ [0, 1]2 there is a circle C ∈ C centered at x, how many pairs of circles can be tangent to each other? We will make this question precise and show how certain quantitative bounds on circle tangencies give us the desired bounds on Mf. First, we shall decompose f into a collection of diadic pieces using a Haar wavelet basis. We will write

∆kf = E[f|Dk] − E[f|Dk+1], (9)

−k where Dk is the σ–algebra generated by diadic squares of length 2 . Thus −k the function E[f|Dk] is constant on each diadic square of side length 2 , and the value of E[f|Dk] on each square is equal to the average of f on that square. We will similarly decompose the measure σ:

∞ X k−1 σ = σ0 + 2 σk, (10) k=1 where σ = χ , 0 1|y|≤2 (11) σk = χ1≤|y|≤1+2−k − χ1≤|y|≤1+2−k+1 .

In order to control kMfkp, it suﬃces to get suﬃciently good control on At(∆kf) for each k. Once this control has been obtained, we can replace P the maximum supt Atf by a sum k supt At(∆kf), and this supremum can in turn be replaced by a sum over an appropriately chosen sequences of t’s.

97 Then, by summing carefully we use the theory of square functions (see i.e. [5]) to obtain the desired Lp bounds on Mf. In order to make this work, we will need to establish the bound

−α(p)s sup At(∆kf) . 2 k∆kfkp . (12) t∼2−(k+s) p

For all choices of k, s ≥ 0. We will prove (12) in the case where k = 0 and f is supported in the unit square. The general case can be recovered by scaling arguments. Equation (12) would follow if for some α > 0 we could establish the bound Z sup ∆ f(x + ty)σ (y)dy 2−k(1+α) kfk . (13) 0 k . p t∼1 p

We will control the RHS of (13) through a certain dualization process. R The operator f 7→ supt∼1 ∆0f(x + ty)σk(y)dy is not linear, and this will make it difficult to find a dual operator. To fix this problem, we shall define a radius function t(x), and we shall consider the expression Z

∆0f(x + t(x)y)σk(y)dy . (14) p

−k(1+α) If we can show that (14) . 2 kfkp , where the implicit constant doesn’t depend on the choice of function t(x), then we will have established (13). Let Vx be the measure with Radon-Nikodym derivative

Vx(y) = χ −1 − χ −1 −1 , (15) dµ t(x)<|x−y|

By duality arguments and Marcinewitz interpolation, it suﬃces to establish the bound Z −1−α 1/q Vx(y)dx . n |Ω| (17) q Ω Ly for some α > 0, and for all measurable sets Ω and all 1 < q < 2.

98 First, we have the following two bounds: Z −1 Vx(y)dx . n |Ω|, (18) 1 Ω Ly Z −1 1/2 Vx(y)dx . n log n|Ω| . (19) 2 Ω Ly Equation (18) follows from Fubini’s theorem, while (19) follows from some general results from [3], which we will not describe here. Interpolating between (18) and (19) establishes (17) for α = 0, but we need to obtain (17) for some α > 0. In order to do this, we will need to improve either (18) or (19). Neither of these bounds can be improved directly, but instead we have the following lemma:

Lemma 3. Let Vx(y) and Ω be as deﬁned above. There exists an absolute constant > 0 and a decomposition Ω = Ω1 t Ω2 such that Z −1− Vx(y)dx . n |Ω1|, (20) Ω 1 1 Ly Z −1− 1/2 Vx(y)dx . n |Ω2| . (21) Ω 2 2 Ly Lemma 3 gives us (17), which in turn establishes Theorem 1 (in the case where Γ is the unit circle). Proof of Lemma 3. Lemma 3 contains some intricate geometric arguments, so we shall only give a general sketch of the main ideas involved. −1 For each x ∈ Ω, let Ex be the annulus of thickness n centered at x of radius t(x). Let

−1+ Ωx = {y ∈ Ω: Ey is (n )–tangent to Ex}. (22) Fix some value of δ that we will choose later. Using the greedy algorithm, select a sequence of points x1, . . . , xJ ∈ Ω such that −1+δ |Ωx1 | > n , −1+δ |Ωx2 \Ωx1 | > n , . . (23) [ −1+δ |ΩxJ \ Ωxj | > n . j

99 Let [ Ω1 = Ωx , j (24) Ω2 = Ω\Ω1. We will ﬁrst establish (21). Let

D = {(x, y) ∈ Ω2 × Ω2 : y ∈ Ωx}. (25)

D is the set of points x, y where Ex and Ey are almost tangent. Ω2 has been constructed so that D is small. We write Z Z 2 Vxdx dy Ω2 Z

= hVx1 ,Vx2 idx1dx2 (x1,x2)∈Ω2×Ω2 Z Z

= hVx1 ,Vx2 idx1dx2 + hVx1 ,Vx2 idx1dx2 (26) D Ω2×Ω2\D (27) The ﬁrst term of (26) is small because the set D is small, while the second term is small because of the following heuristic:

Heuristic 4. If two annuli Ex1 and Ex2 are far from being tangent, then hVx1 ,Vx2 i is very small:

The diﬀerence between the areas of the regions F++ ∪ F−− and F+− ∪ F−+ is small. We will now establish (20). Deﬁne 0 Ω1 = Ωx1 , 0 Ω2 = Ωx1 \Ωx1 , 0 Ω3 = Ωx3 \(Ωx1 ∪ Ωx2 ), . .

100 It suﬃces to show that

Z −1− 0 Vx(y)dx . n |Ωj| (28) 0 Ωj 1 Ly for each index j. After a translation and scaling, we can assume that xj = 0 −1 and t(xj) = 1, i.e. Exj is the n /2 neighborhood of the unit circle. After 0 diadic pigeonholing (and a corresponding reﬁnement of Ωj by a factor of log n), we can also assume that there exists a diameter d such that |x| ∼ d 0 0 for all x ∈ Ωj. After a suitable modiﬁcation of the functions {Vx : x ∈ Ωj}, we establish (28) using the Cauchy-Schwartz inequality and a suitable bound 2 R on the L –norm of 0 Vx(y)dx. Again, we need to show that not too many Ωj tangencies can occur amongst the annuli from

0 {Ex : x ∈ Ωj}. (29)

In general, two circles (or annuli) can be tangent to each other even if their centers are far apart from each other. However, the annuli in (29) are all almost tangent to the unit circle, and thus we can apply the following heuristic: Heuristic 5. Let C(x, r),C(x0, r0) be circles, both of which contain a common point z, which lies slightly outside the unit circle. Suppose C(x, r) and C(x0, r0) are each nearly tangent to the unit circle. Then the extent to which C(x, r) and C(x0, r0) are tangent to each other is controlled by |x − x0|. Using this heuristic, we can control the number of almost tangencies in (29), and this in turn allows us to establish (28).

References

[1] J. Bourgain. On the spherical maximal function in the plane. Preprint. IHES, 1985.

[2] J. Bourgain. Averages in the plane over convex curves and maximal operators. J. Anal. Math. 47(1):69–85, 1986.

[3] J. Bourgain. On high dimensional maximal functions associated to convex bodies. Am. J. Math. 108(6):1467–1476, 1986.

101 [4] W. Schlag. Lp → Lq estimates for the circular maximal function. Ph.D. Thesis. California Institute of Technology, 1996.

[5] E. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton NJ, 1993.

Joshua Zahl, UCLA email: [email protected]

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