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Harmonic Analysis, Geometric Measure Theory and Additive Combinatorics

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Harmonic Analysis, Geometric Measure Theory and Additive Combinatorics Harmonic Analysis, Geometric Measure Theory and Additive Combinatorics 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 appli- cations 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 appli- cations 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 op- erators 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 2 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 2 E. In particular, given any δ > 0, we may choose an interval I = (x − , x + ) such that jE \ Ij ≥ (1 − δ)jIj. 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 1 infinite sets are universal: for instance, if A = fangn=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 1 remains open for A = f2 gn=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 2 [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 = f0; 1; 2g 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" fx; x + r; y; y + rg 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 f0; 1; 2g 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 proba- bilistic measure µ with the following properties: α (A) µ([x; x + ]) ≤ C1 for all 0 < ≤ 1, −B − β (B) jµb(k)j ≤ C2(1 − α) jkj 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 \ f1;:::;Ng) lim > 0; (1) N!1 N then A must contain a non-trivial 3-term arithmetic progression. Szemer´edi'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 natu- ral 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, de- pending on the size of A, then A must contain 3-term arithmetic progressions even if its asymptotic density is 0.
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