Arithmetic Structures in Small Subsets of Euclidean Space

Home , Approximate identity, Geometric measure theory

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Marc Carnovale, M.S.

Graduate Program in Mathematics

The Ohio State University

2019

Dissertation Committee:

Prof. Vitaly Bergelson, Advisor Prof. Alexander Leibman Prof. Krystal Taylor c Copyright by

Marc Carnovale

2019 Abstract

In this thesis we extend techniques from additive combinatorics to the setting of harmonic analysis and geometric measure theory. We focus on studying the distribution of three- term arithmetic progressions (3APs) within the supports of singular measures in Euclidean space. In Chapter 2 we prove a relativized version of Roth’s theorem on existence of 3APs for positive measure subsets of pseudorandom measures on R and show that a positive measure of points are the basepoints for three-term arithmetic progressions within these measures’ supports. In Chapter 3 we combine Mattila’s approach to the Falconer distance

d conjecture with Green and Tao’s arithmetic regularity lemma to show that measures on R with suﬃciently small Fourier transform as measured by an Lp-norm have supports with an abundance of three-term arithmetic progressions of various step-sizes. In Chapter 4

d we develop a novel regularity lemma to show that measures on R with sufficiently large dimension, as measured by a gauge function, must either contain non-trivial three-term arithmetic progressions in their supports or else be structured in a specific quantitative manner, which can be qualitatively described as, at infinitely many scales, placing a large amount of mass on at least two distinct cosets of a long arithmetic progression.

ii Aknowlegements

First and foremost, I would like to thank my Ph.D. advisor Vitaly Bergelson, who set me on the road many years ago. His constant support and generosity have had a greater impact on me than I could ever express.

In addition to Vitaly Bergelson, who taught me so much of what I know about mathematics, I would especially like to thank my M.Sc. advisors IzabellaLaba and Malabika

Pramanik who taught me much of the rest.

Many thanks also to my thesis defense committee members Alexander Leibman and

Krystal Taylor for their time and encouragement.

Finally I would like to thank my parents, family, and friends, who have always been there for me and provided endless support.

iii Vita

2010 ...... B.S. Mathematics Ohio State University. 2013 ...... M.S. Mathematics University of British Columbia. 2013-present ...... Graduate Teaching Associate, Ohio State University.

Publications

Research Publications

M. Carnovale “A relative Roth theorem in dense subsets of pseudorandom fractals.” Online Journal of Analytic Combinatorics, 27–55, June. 2015.

Fields of Study

Major Field: Mathematics

iv Table of Contents

Page

Abstract ...... ii

Vita...... iv

List of Tables ...... vii

1. Introduction ...... 1

1.1 Notation ...... 2 1.2 A selected history of additive combinatorics ...... 4 1.3 A selected history of geometric measure theory ...... 6 1.4 A selected history of harmonic analysis ...... 9 1.4.1 The Restriction Conjecture ...... 10 1.4.2 The multilinear Hilbert transform ...... 12 1.5 Content of this thesis...... 14

2. A relative Roth theorem in dense subsets of sparse pseudorandom fractals . . . 16

2.1 Notation and approach ...... 19 2.2 Existence of the restricted measure on 3APs ...... 21 2.3 The Main Estimate : Proof of Lemma 2.3.2 ...... 25 2.4 3APs in dense subsets : Proof of Theorem 2.0.1 ...... 32 2.5 Existence of the measure on 3APs ...... 33 2.6 Further Remarks ...... 40

3. On the number of 3APs in fractal subsets of Euclidean Space ...... 42

3.0.1 Integer sets ...... 43 3.0.2 Fractal sets ...... 44 3.0.3 Finite ﬁelds ...... 50 3.1 Integer results ...... 51 3.1.1 3AP counts of L2 functions ...... 58

v 3.2 Fractal results ...... 62 3.2.1 Proof of Theorem 3.0.7 ...... 63 3.2.2 Proof of Theorem 3.0.8 ...... 73 3.3 Finite ﬁeld results ...... 77 3.3.1 Proof of Corollary 3.0.11 ...... 77 3.3.2 Gauss sums ...... 78 3.3.3 Circular bounds and the proof of Theorem 3.0.14 ...... 82 3.3.4 Decay and the proof of Theorem 3.0.12 ...... 83

4. 3APs in dense subsets of Euclidean Space ...... 86

4.1 Introduction ...... 86 4.2 Large gauges without 3APs ...... 88 4.3 Almost flat fractals ...... 89 4.4 Counting 3APs in almost-flat fractals ...... 90 4.4.1 Counting 3APs between three progressions ...... 90 4.4.2 The counting lemma ...... 95 4.5 An (L1, U 2) regularity lemma ...... 98 4.5.1 The regularity lemma ...... 98 4.5.2 Bohr sets ...... 104 4.6 Decomposing a large fractal into an almost-flat portion and a small pseudorandom error ...... 106 4.7 Counting 3APs in large fractals ...... 108

Bibliography ...... 109

vi List of Tables

Table Page

1.1 Comparison of discrete versus fractal results on 3APs...... 15

vii Chapter 1: Introduction

This thesis explores applications of additive combinatorics to Euclidean harmonic analysis and geometric measure theory through the lens of one very speciﬁc family of structures: arithmetic progressions. We are particularly interested in the study of their presence within subsets of Hausdorﬀ dimension less than d of d-dimensional Euclidean space (we will refer to such subsets as ‘fractals’). Such a study was initiated by the seminal 2009 paper ofLaba and Pramanik [35] (see Theorem 1.4.1 in Section 1.4 below).

There are several reasons such a study may be of interest. First, a number of recent advances in Harmonic Analysis have seen connections to questions about the distribution of arithmetic progressions. Second, it is natural to count the number of k-term arithmetic progressions (kAPs for short) in a set A with characteristic function 1A by introducing the multilinear functional

ZZ Λk(f0, . . . , fk−1) := f0(x)f1(x − r) . . . fk−1(x + (k − 1)r) dx dr

and applying it to the tuple (f0, . . . , fk−1) = (1A,..., 1A); it is the consequence of monu- mental advances in combinatorial number theory, additive combinatorics, and ergodic theory that in many contexts these multilinear funtionals are fairly well-understood compared to two to three decades ago. Finally, many of the tools being developed in additive combinatorics to understand discrete problems are of a ﬁnitistic Fourier-analytic nature but largely novel to Euclidean harmonic analysis, and extending them to this setting is often non-trivial.

1 It is our belief that important future advances in Euclidean harmonic analysis will continue

to beneﬁt from tools developed in additive combinatorics.

1.1 Notation

Given quantities A and B which may depend on auxiliary parameters ~s = hs1, . . . , ski

we write

A . B

to indicate the existence of a ﬁnite positive constant C (independent of these parameters

unless otherwise noted) such that

A ≤ CB.

uniformly in these parameters.

N Similarly, when A = A , B = B , and lim (1+As) = 0 for some power N > 1, we s s s → ∞ Bs will write A B.

We use also the notation A & B when B . A and A B when B A, and we write

A ≈ B when both A . B and B . A hold.

Given an integer N ∈ N, write [N] to denote the set

[N] := {n ∈ N : 0 ≤ n < N} ;

(note that this diﬀers from standard usage where [N] denotes the integers from 1 to N).

d Deﬁne the Fourier transform of a function f or Borel measure µ on R by Z fˆ(ξ) = f(x)e−2πiξ·x dx, d R Z µ(ξ) = e−2πiξ·x dµ(x) b d R d for ξ ∈ R .

2 For a function f on ZN := Z /N Z deﬁne for ξ ∈ ZN

1 X fˆ(ξ) = f(x)e−2πinξ/N . N x∈ZN ˆ For a function f :[N] → C, we deﬁne f(ξ) via ﬁrst identifying f with a function on Zp

for p a prime between 3N and 6N via the embedding [N] 7→ Zp, n 7→ (n mod p) ∈ Zp and then taking the Fourier transform on this group. That such a prime, p, always exists is

guaranteed by Bertrand’s postulate.

Given functions f, g : G 7→ C on an abelian group, G with Haar measure m, we denote convolution with an asterisk:

Z f ∗ g(x) = f(x − y)g(y) dm(y), G

and extend this to ﬁnite Borel measures in the usual way, R f dµ∗ν := R f(x+y) dµ(x) dν(y).

We will denote the Lebesgue measure of a set, E, by |E|.

We will at times write Lˆq to refer to the space of functions whose Fourier transforms

possess ﬁnite Lq norms.

d Denote the ball of radius r centered at a point x in R by B(x, r), and let σ denote the surface measure on the unit sphere.

In this thesis, we will very often be interested in positive Borel measures µ on [0, 1]d

satisfying the following Hausdorﬀ dimension and Fourier decay conditions

α (a) µ(B(x, r)) ≤ CH r

and

−β/2 (b) |µb(ξ)| ≤ CF (1 + |ξ|)

for some positive constants CH and CF .

We will refer back to conditions (a) and (b) frequently.

3 1.2 A selected history of additive combinatorics

The study of the distribution of kAPs within various subsets of abelian groups is one of the central questions in combinatorial number theory and additive combinatorics, dating back to a 1927 theorem of van der Waerden, [53], stating that any parition of the natural numbers into ﬁnitely many subsets must contain a cell which contains arbitrarily long arithmetic progressions, and a later conjecture of Erd˝osand Tur´an,[14], asking whether any large enough subset, A, of the natural numbers must contain arbitrarily long arithmetic

progressions. Here by large enough is meant that the upper density of the set be positive,

that is

|A ∩ [N]| d(A) := lim sup > 0. (1.1) N → ∞ N

Erd˝osis also known for another conjecture, namely that to guarantee kAPs in a set A

for arbitrary k it is suﬃcent that

X 1 = ∞. (1.2) n n∈A

It is easily seen that (1.2) is strictly stronger than (1.1); in fact, (1.2) is roughly equivalent to asking whether

|A ∩ [N]| 1 d (A) := (1.3) [N] N & log N

(we refer the reader to [23] for a precise statement of (1.3)).

By the Prime Number Theorem, the density d[N](P) of the set of primes, P, is approx-

1 imately log N , and so an aﬃrmative answer to the question of whether, for any k ∈ N,

1 d[N](A) & log N for suﬃciently large N implies that the set A contains a (non-trivial) kAP would in turn imply that statement for the primes, answering an older question of Erd˝oson

whether the primes contain arbitrarily long arithmetic progressions.

4 Using special properties of the prime numbers, in 1939 van der Corput showed that the primes do indeed contain inﬁnitely many 3APs ([52]) but this proof does not extend readily to other sets. The full question of whether the primes contain arbitrarily long arithmetic progressions went unanswered until the 2008 breakthrough paper [26] of Green and Tao.

In 1954 Roth ([45]) gave a proof that (1.1) implies that the set A contains (non-trivial)

3APs, and his proof gives something a little more quantitative than that.

Theorem 1.2.1 (Roth’s Theorem). Let A ⊂ [N] and suppose that

1 d (A) . [N] & log log N

Then provided that N is suﬃciently large, the set A contains non-trivial three-term arithmetic progressions.

A key feature of Roth’s proof is that it does not directly show that the original set contains

3APs, but rather proceeds by reﬁning this set into a subset A0 which is either “structured” in the sense that A0 is itself a long arithmetic progression, or “pseudorandom” in the sense that 2 |A0| 0 sup 1 0 (ξ) < (which may be shown to roughly mean that A is approximately ξ6=0 cA N evenly distributed over the long enough sub-progressions of [N].1). In both cases, it is easy to show that the subset contains 3APs, so the main work is in showing this dichotomy that any set must exhibit either “structured” or “pseudorandom” behaviour.

In 2005 Green ([25]) extended Roth’s theorem (as a theorem about the natural numbers

N) to the primes by exploiting deep results that the primes are especially pseudorandom modulo trivial congruence properties. Chapter 2 takes particular inspiration from this result.

In 1975, Szemer´ediextended Roth’s theorem by proving the following.

Theorem 1.2.2 (Szemer´edi’s Theorem, [47]). Let k ∈ N. If A ⊂ [N], |A|/N ≥ δ, and N is suﬃciently large depending on k and δ, then A contains kAPs.

1In fact, in Roth’s proof of his theorem as well as in a majority of other proofs, one does not obtain this decomposition in a single step but rather as a result of successive reﬁnements of the original set.

5 Many consider this one of the most signiﬁcant results of 20th century mathematics. A

second proof was provided by Furstenberg, using ergodic methods, in 1977 ([18]). This led

to many siniﬁcant extensions, some of which remain inaccessible via non-ergodic methods

(c.f. [19],[20],[3],[4]).

1.3 A selected history of geometric measure theory

Geometric measure theory studies the geometry of sets through the viewpoint of measures

supported on those sets. Perhaps the most fundamental geometric data accessible via such

d measures is the Hausdorff dimension, dimH E, of a set E ⊂ R . The usual definition of Hausdorff dimesion goes through certain coverings and outer

measures as follows.

d Let R be the semi-ring of sets generated by the balls in R . Let s ∈ [0, d].

0 s 0 Given a ball B(x, r), set ms(B(x, r)) = r and extend ms to R via additivity. For a given δ > 0, let R0 denote the sub-semi-ring R0 ⊂ R consisting of those elements of

s R with diameter less than δ, and deﬁne Hδ to be the measure generated by the pre-measure

0 0 ms on R .

s It is easy to see that for any set E, Hδ(E) is monotonic increasing as δ ↓ 0.

Definition 1.3.1. Define Hs, the s-dimensional Hausdorff measure, to be given by Hs(E) :=

s s supδ>0 Hδ(E) = limδ↓0 Hδ(E).

We also deﬁne the Hausdorﬀ dimension of a measure µ as

α dimH µ = sup {α ∈ (0, d): ∃ C such that µ(B(x, r)) ≤ Cr for µ − a.e.x} .

It is a consequence of the following theorem of Frostman that the Hausdorﬀ dimension

of a set is a geometric measure-theoretic quantity.

6 d Theorem (Frostman’s lemma, c.f. [40] Theorem 8.8.). Let E ⊂ R be a Borel set. Then Hs (E) > 0 if and only if there exists a positive Borel measure, µ, supported on E such that

µ (B(x, r)) ≤ rs

d for µ-a.e. x ∈ R .

In fact, there is a weaker notion of dimension for Euclidean measures than that guaranteed by Frostman’s Lemma which is more analytically tractable and often suﬃcient in applications. We give the deﬁnition below.

Given a real number s and a distribution f, deﬁne the Sobolev norm kfk d by W 2,s(R )

Z 2 1/2 s kfk d := f(ξ) (1 + |ξ|) dξ . W 2,s(R ) b

We are particularly interested in the (negative exponent) Sobolev norm

Z 2 2 −(d−α) kfk 2,−(d−α) d = f(ξ) (1 + |ξ|) dξ W (R ) b for α ∈ (0, d).

We now deﬁne

2 I (µ) := kµk 2,−(d−α) d , α W (R ) usually termed the α-dimensional energy of µ.

It turns out that Iα has a more direct spatial representation, stemming from the (distribu-

−(d−α) −α tional) Fourier transform cd,α|ξ| of the function x 7→ |x| (where cd,α is an appropriate constant),

ZZ 1 I (µ) = c dµ(x) dµ(y). α d,α |x − y|α

One then deﬁnes the capacitary dimension, dimcap(µ), as

dimcap µ = sup {α ∈ (0, d): Iα(µ) < ∞}

7 with the convention that the supremum of the empty set is zero.

Note that since the integral R 1 dm (x) < ∞ for all > 0 (and is inﬁnite for B(0,1) |x|d− d

= 0), the capacitary dimension of Lebesgue measure md is d.

It is straightforward that if dimH µ = α, then dimcap µ ≥ α.

If dimcap µ > α for some α ∈ (0, d) then it can be shown that one may select a new measure, ν, supported on a subset of supp µ such that dimH ν > α. Thus Frostman’s lemma implies the variant of itself where dimH µ is replaced by dimcap µ.

There is another quantity often referred to as a dimension and which is of particular interest within harmonic analysis. We noted that dimH µ > α implies that dimcap µ > α, and we have that

0 dimcap µ = sup {α ∈ (0, d): Iα0 (µ) < ∞} where Z 2 −α0 Iα0 (µ) = |µ(ξ)| |ξ| dξ. d b R −α/2 Thus, informally, |µb| must decay as |ξ| → ∞ like |ξ| on average. If for some β > 0, −β/2 |µb(ξ)| . |ξ| for all (large enough) ξ, then certainly dimcap µ ≥ β, though it is not necessary that there be any such β > 0 even when µ is of full Hausdorﬀ dimension. Nonetheless, it is useful to deﬁne

n −β/2o dimF µ := sup β ∈ (0, d): |µb(ξ)| . |ξ| ,

referred to as the Fourier dimension of µ.

Many basic questions in geometric measure theory take the form of asking whether large-

ness of the dimension of a set is suﬃcient to guarantee structure of the set under question.

One of the most important unresolved questions in the ﬁeld is of this form, where we make

d “largeness” precise by asking, given E ⊂ R , that dimH E > d/2, and make structure precise

8 by asking that the set

∆(E) := {r = |x − y| : x, y ∈ E} have positive Lebesgue measure.

In terms of measures, this may be restated as

d Conjecture 1.3.2 (Falconer, [16]). Suppose that µ is a probability measure on R and that

α µ(B(x, r)) . r for some α > d/2. Then

|{r : there exist x, y ∈ supp(µ) such that |x − y| = r}| > 0.

In Chapter 3 we adopt the viewpoint that the Falconer distance conjecture asks a question about the distribution of 2APs in supp µ under the assumption that dimH µ > d/2. If one replaces dimH by dimF , then it is an observation of Mattila ([39]) that the conclusion of the

Falconer distance conjecture holds true.

1.4 A selected history of harmonic analysis

In their 2009 paper [35],Labaand Pramanik proved:

Theorem 1.4.1 (Laba-Pramanik, 2009). Suppose that µ is a probability measure on [0, 1] satisfying2 conditions (a) and (b) for some β > 2/3 and α suﬃciently close to 1 dependng on β and the constants CH ,CF appearing in (a) and (b). Then supp µ contains non-trivial

3APs.

There are a number of results in harmonic analysis where the arithmetic structure inﬂu- ences the analytic or geometric/combinatorial structure. In this section we discuss a few of them, and discuss how each relates, also, to the kind of 3AP counts ﬁrst studied by Theorem

1.4.1.

2See Section 1.1 for the statements of conditions (a) and (b).

9 1.4.1 The Restriction Conjecture

One of the most signiﬁcant areas of research in Euclidean harmonic analysis is the study

of the (Fourier) Restriction Conjecture.

The Restriction Conjecture3 relates the concentration of the energy of the Fourier trans-

form of a function f, as measured by some Lbq(σ) norm, fˆ , to its mass (as measured Lq(σ) p by some L norm, kfkLp ), by asking whether

ˆ f . kfkLp( d) . Lq(σ) R

It is not hard to observe that the curvature of the sphere plays a signiﬁcant role in the

problem. For instance, instead of σ consider the Lebesgue measure mV supported on a ﬂat

d d aﬃne subspace V ⊂ R . Then given f on R , ﬁrst, we have

f\ dλ d ≤ kfk q0 R d L (λ d ) Lq(R ) R

for q ≥ 2 (this is the classical Hausdorﬀ-Young inequality, and it is known to be sharp in

d d that no further Lp(R ) − Lq(R )-inequality is true in general); and second, it is easy to see that

f\ dλV 6 kfk p d . L (mV ) Lq(R )

d for any (p, q) 6= (1, ∞) for any proper aﬃne subspace V 0 (R .

On the other hand, if one replaces the ﬂat measure mV with the surface measure mS

d where S is any hypersurface of non-vanishing Gaussian curvature in R , then one has the remarkable inequality

f\ dmS ≤ kfk 2 , d L (mS ) Lq(R )

3Stein posed the Restriction Conjecture in the 1970’s, asking whether the following is true.

p d 2d Conjecture 1.4.2 (Restriction Conjecture, Stein). Let f ∈ L (R ) for some p ∈ [1, d+1 ) and suppose that d−1 0 0 p q ≤ d+1 p where p = p−1 . Then

ˆ f . kfkLp( d) . Lq (σ) R

10 d+1 d−1 for q ≥ 2 2−1 , first proven for the d-dimensional unit sphere, S = S , by Tomas ([51]) for d+1 d+1 ˆ q > 2 2−1 and by Stein for the endpoint q = 2 2−1 . This inequality shows that although f is only initially defined Lebesgue-almost-everywhere, it has a well-defined restriction to the

(Lebesgue-measure zero) sphere. This theorem was later generalized by Greenleaf ([28]) to the case of general hypersurfaces of non-vanishing Gaussian curvature.

For a long time the phenomenon of Fourier Restriction was perceived to rely on the geometry of certain smooth manifolds. Then around the year 2000, it was noticed that the structure of a smooth manifold doesn’t play any vital role in the estimate, and in fact the

Stein-Tomas restriction theorem holds for any measure µ satisfying appropriate Hausdorﬀ dimension and Fourier decay bounds (those same ones encapsulated by conditions (a) and

(b) stated in Section 1.1).

Theorem 1.4.3 (Mitsis [42], Mockenhaupt [43], endpoint Lq bound by Bak-Seeger [1]). Let

d µ be a probability measure on R satisfying (a) and (b) for some CH and CF ∈ (0, ∞).

d−α+β 2 Then given any q ≥ 2 2(d−α)+β and any f in L (µ),

f dµ ≤ kfk 2 . d d L (µ) Lq(R )

In [43], Mockenhaupt remarks that (bounds on) the number of long progressions in the

set may play the role of curvature hypotheses in the fractal setting, and this was made

more precise by the following result ofLabaand Hambrook, showing sharpness of the Mitis-

Mockenhaupt-Bak-Seeger result by building a measure whose support contains many long

progressions at small scales.

Theorem 1.4.4 (Laba-Hambrook [29]). For every α0 ∈ (0, 1) there exists an α > α0 such

4 that for all p ∈ [1, α − 2), for all β < α there exists a measure µ satisfying (a) and (b), and

11 a sequence of functions fl, such that

f[l dµ p L (R) → ∞. kflkL2(dµ)

We note also that the Mitsis-Mockenhaupt-Bak-Seeger Stein-Tomas restriction theorem

has the same hypotheses as theLaba-Pramanik 3AP result Theorem 1.4.1 - namely the

conditions (a) and (b). We can rephrase this as saying that the hypotheses which lead

to the conclusion of Theorem 1.4.3 are connected to those which allow for good estimates

on 3AP counts within fractals. In the other direction, we have theLaba-Hambrook result

demonstrating sharpness of the Mitsis-Mockenhaupt restriction theorem in the case of certain

measures with very bad kAP counts. Given these two observations, it is natural to conjecture

that good control on the number of kAPs for k much larger than 3 might allow for L2 − Lp

restriction theorems for exponents p below that guaranteed by Theorem 1.4.3. Thus it seems

desireable to better understand arithmetic behaviour of fractals associated to the counts of

the arithmetic progressions in their support.

1.4.2 The multilinear Hilbert transform

Another of the outstanding open questions in harmonic analysis is that of boundedness

of the multilinear Hilbert transform. The Hilbert transform is deﬁned initially for Schartz

dt R 1 functions f : R → C (and treating t as a tempered distribution) as Hf(x) = f(x − t) t dt,

R 1 and the bilinear Hilbert transform as H(f, g)(x) = f(x−t)g(x+t) t dt. Boundedness of the

Hilbert transform on Lp(R) for p > 1 was established by by Calder´onand Zygmund in 1952 in their seminal study of singular integrals, ([7]). Boundedness of the Bilinear Hilbert transform from Lp1 ( ) × Lp2 ( ) to Lq( ), for 1 + 1 = 1 when q ∈ ( 2 , ∞), was proven by Lacey and R R R p1 p2 q 3 Thiele in the 90’s [36], [37], answering a decades old question of Calder´on.More generally,

R Qk dt p given fi : R → R, i = 1, . . . , k, one deﬁnes H(f1, . . . , fk)(x) = i=1 fi(x − it) t . L - boundedness of the trilinear and higher order Hilbert transforms remains entirely open. Tao

12 recently used tools from additive combinatorics4 in order to obtain estimates for truncations5 of the mulitlinear Hilbert transform which blow up logarithmically more slowly than the trivial estimates available via Holder’s inequality ([48]).

One may associate a multilinear form to the Lp norm of the multilinear Hilbert transform via Lp duality as follows:

Z kH(f1, . . . , fk−1)kLp = sup g(x)H(f1, . . . , fk−1)(x) dx kgk 0 Lp =1 ZZ dt = g(x)f (x − t)f (x − 2t) . . . f (x − (k − 1)t) dx . 1 2 k−1 t

If one considers a modiﬁcation of the tri-linear form associated to the norm of the bilinear

1 Hilbert transform, in which the singular measure t dt is replaced by the Lebesgue measure dt, then one obtains the 3AP-counting functional

ZZ Λ3(f0, f1, f2) := f0(x)f1(x − t)f2(x − 2t) dx dt, (1.4) and so perhaps it is not so surprising that recent advancements on the study of multilinear

Hilbert transforms have used some of the same technology as that appearing in the study of

Λk(f0, . . . , fk−1) and the distribution of arithmetic progressions.

If one replaces the functions fi in (1.4) with φ ∗ µ where φ is an approximate identity and µ is a probabilty measure satisfying (a) and (b), then this is the primary object of study to obtain existence of 3APs within the support of the fractal µ, as in, e.g., Theorem 1.4.1.

This is one manner in which the study of 3APs within singular sets may be seen as a model for importing tools from additive combinatorics to a harmonic analysis setting.

4In particular, the Inverse Conjecture for the Gowers norms in the form of the Arithmetic Regularity Lemma. 5Given , deﬁne k Z Y dt H(f1, . . . , fk)(x) := fi(x − it) . c t B(0,) i=1 The bounds in [48] blow up, as ↓ 0, more slowly than would be implied by Holder’s inequality by a logarithmic factor in .

13 1.5 Content of this thesis.

Our primary results in this thesis focus on a delicate analysis of the 3AP counts within

the support of a measure µ, taking the study instigated by [35] as an excellent model case

for applying the tools of additive combinatorics in a Euclidean harmonic analysis setting.

In Chapter 2 we extend the main result ofLaba and Pramanik’s paper “Arithmetic

progressions in sets of fractional dimension” ([35]) to obtain a “relative” version of their

theorem along the lines of the 3AP version of the “relative Szemer´editheorem” of Conlon,

Fox, and Zhao [10] (this relative Szemer´editheorem, extending work of [25] and [26], showed

that dense subsets of appropriately pseudorandom sets, not just dense subsets of the full

space, contain arbitrarily long arithmetic progressions). In [35], the authors proved that any

measure µ on [0, 1] with Hausdorﬀ dimension α ∈ (1 − 0, 1) (here 0 is a small constant) large enough depending on its Fourier dimension β ∈ (2/3, α] and the parameters CH ,CF

of (a) and (b) contains in its support three-term arithmetic progressions (3APs). Here, we

adapt an approach introduced by Green in “Roth’s Theorem in the Primes”, [25], to show

for any δ > 0 that if α is large enough depending on δ (and the other data in the hypotheses)

then µ gives positive measure to the (basepoints of the) non-trivial 3APs contained within

any set A for which µ(A) > δ.

In Chapter 3, we use techniques introduced by Mattila in [40] to study the Falconer

distance conjecture, to explore conditions which guarantee largeness (in terms of bounded

L2 density/Lebesgue measure and Hausdorﬀ measure) of the set of lengths of step-sizes of

3APs which occur within fractal sets. Our main result is a version ofLabaand Pramanik’s

theorem in [35] that relies only on an assumption of a lower bound, δ, on the mass of the

measure µ together with an upper bound, M on the Lq norm of its Fourier transform for some q ∈ (2, 3] depending on the parameters δ and M. In fact, we show that conditions (a) and (b)

14 Comparison of Discrete vs. Fractal Settings Discrete sets on [N] Singular measures on [0, 1]d Roth’s Theorem Theorem 4.1.3 Sparse Pseudorandom Roth’s Theorem Laba-PramanikTheorem 1.4.1 Green’s Relative Roth Theorem Theorem 2.0.1

Table 1.1: Comparison of discrete versus fractal results on 3APs.

imply our assumption on the Lˆq norm of µ, so that out theorem implies theLaba-Pramanik

as a special case. We also obtain analogous statements in the discrete setting.

In Chapter 4 we use a novel regularity lemma to show that any set E ⊂ [0, 1] satisfying

the hypothesis that E have positive generalized Hausdorﬀ measure Hh(E) > 0 with respect

to a suﬃciently large gauge function, h, must either contain non-trivial 3APs or possess a

speciﬁc structure on its support. The proof uses the regularity lemma to replace the measure

µ with a new measure µe with approximatley the same 3AP count behaviour but which is “structured” in the sense that it looks like a measure built via a Cantor-type construction.

We then use this structure to obtain strong enough counts on the number of 3APs within

the support of µe, as well as their distribution, to guarantee that the set E must contain non-trivial 3APs as well.

This allows us to complete Table 1.1 comparing what is known in the discrete versus

fractal settings and how these results relate to one another.

15 Chapter 2: A relative Roth theorem in dense subsets of sparse pseudorandom fractals

This chapter is primarily work published in [8].

In this chapter we employ the technology of restriction/transference results in additive

combinatorics developed in [25] for the prime numbers to extend the result, Theorem 1.4.1, of

Labaand Pramanik [35] demonstrating three-term arithmetic progressions in certain fractal

sets.

We will be concerned with a measure µ on T satisfying the following conditions for

appropriate α, β, CH ,CF in the case that d = 1

α d (a) |µ|(B(x, r)) ≤ CH r for all x ∈ T

− β d 6 (b) |µb(ξ)| ≤ CF (1 + |ξ|) 2 for all ξ ∈ Z .

In [35], it was shown that a measure µ satisfying (a) and (b) for α suﬃciently close

to 1 depending on β > 2/3, CH , and CF must contain in its support 3-term arithmetic

progressions. Consequently, any closed set supporting a measure satisfying (a) and (b) with

appropriate constants must contain 3APs. In the present thesis, we show that in fact, dense

subsets of such sets still contain 3APs, in particular providing a condition for a merely

measurable set to contain progressions. Our main result is

6The largest α and β ∈ (0, 1) for which (a) and b hold are referred to as the Hausdorﬀ and Fourier dimensions of the measure µ. For more on these, see, e.g., [40].

16 Theorem 2.0.1. Let δ > 0, and suppose that the probability measure µ on T satisﬁes (b)

1 with β > 2 and (a) with α suﬃciently close to 1 depending on β, δ, and the implicit constants

CH and CF in (a) and b.

Then any measurable set A with µ(A) > δ contains 3APs 7 ; indeed, µ gives positive

measure to the set of x such that x, x + r, x + 2r is a non-trivial 3AP contained with A.

Remark 2.0.1. Notice that it follows from the above Theorem that the total measure with

respect to µ of the set of x which are not the basepoint of a 3AP entirely contained within

supp µ is less than δ.

The proof of Proposition 2.4.1 (upon which the proof of the above theorem rests) requires

9−2β that α > 8 . By Remark 2.3.1, in the case that δ = 1 and CH ,CF ≈ 1, it is suﬃcient to take α > 1 − β/6 for Lemma 2.3.2. Thus if β = α in the full-density case, as is nearly-

achieved by Salem sets, then we need only take α = β > 9/10. It is unclear whether the

dependency of α on the parameter δ < 1 may be removed. For details, see Section 2.6.

This result even applied to the full support of the measure is novel in that we show µ

gives positive measure to the basepoints of non-trivial 3APs contained in its support.

Our approach is based on [25], in which it was shown that 3APs are contained in any

set taking up a positive proportion of the prime numbers. There, the main ingredients were

the pseudorandomness of the prime numbers, a restriction theorem for pseudorandom sets

(though in that paper stated only for the prime numbers), and the properties of Bohr sets.

In the present context, all three ingredients are available. Namely, the Fourier decay condi-

tion (b) plays the role of pseudorandomness, we invoke Theorem 2.0.2 to obtain restriction

estimates, and the properties of Bohr sets carry over unchanged to the continuous setting.

7Note that by requiring that supp(µ) ⊂ [1/3, 2/3] ⊂ [0, 1] ≈ T, or equivalently by dilating, we guarantee that the progressions above are genuine progressions when supp(µ) is embedded in R.

17 One reason to be interested in the arithmetic properties of fractional sets is their impli-

cation in the ﬁne analytic behaviour of singular sets.

Recall from Chapter 1 that in 2002 (respectively, 2000) Mitsis [42] (independently Mock-

enhaupt [43]) obtained the following Stein-Tomas type restriction theorem for fractional

sets.8

d Theorem 2.0.2. Let µ be a compactly supported positive measure on R which obeys (a)

2(2d−2α+β) and (b) for some α, β ∈ (0, d). Then for all p ≥ p◦ = p◦(d, α, β) := β , there is a C(p) > 0 such that

9 kfdµdk`p(Zd) ≤ C(p)kfkL2(µ).

As noted by Mockenhaupt [43], obstructions to restriction results for the sphere rely on

arithmetic properties of the surface measure on the sphere (more speciﬁcally, on the existence

of arithmetic progressions in small neighborhoods of pieces of this measure). Recent work

ofLabaand Hambrook [29] made more precise the analogy between arithmetic progressions

in fractional sets and the classical Knapp example showing sharpness of the spherical Stein-

Tomas theorem. They did so by constructing measures µ on R obeying (a) and b which nonetheless contained an abundance of long arithmetic progressions, and demonstrating

that an extension of Theorem 2.0.2 to an improved Lp range fails for such µ via an argument

which makes direct use of the many long arithmetic progressions contained in supp(µ). Thus

an understanding of deep properties of singular measures can be seen to begin with the easier

problem of understanding their arithmetic properties.

This work is particularly motivated by a desire to directly connect to harmonic analysis

a principle in additive combinatorics that a not too small pseudorandom subset of a space

should possess many of the same properties as the full space, and results for subsets of

8Aside from the end point, which was obtained by Bak and Seeger in 2010.

9 In [42] and [43] this theorem was stated with kfdµk p d in place of kfdµk p d . By Lemma 2.3.3, the d L (R ) d ` (Z ) two are equivalent for p ∈ (1, ∞]

18 the entire space should, under suitable hypotheses, hold for subsets of the pseudorandom subspace. The most well-known example of such a phenomenon is the celebrated Green-Tao theorem that the primes contain arbitrarily long arithmetic progressions [26], whose proof entailed ﬁrst demonstrating that the prime numbers behave in a suitably pseudorandom fashion and second developing a “relative Szemer´edi”theorem for subsets of appropriately pseudorandom subsets of the integers. Their result extends ideas present in Green’s earlier

Roth’s Theorem in the Primes [25], from which we take inspiration in the present chapter.

The principle of such transference from a large global space to a nice but sparser subspace can be seen in earlier work, such as in Stein’s Spherical maximal theorem [46], which ﬁnds that the measure on the sphere behaves almost “as well” as the measure on the unit ball for purposes of obtaining a maximal theorem, and similarly in Bourgain’s ergodic theorem along the squares [6]. What many of these results have in common is that they rely on the pseudorandom set being close in a Fourier or spectral sense to the indicator function for the entire space. In [25], this is expressed by a Fourier restriction estimate which underlies the result, but in [26], as further developed in [50], abstracted in [22] and [44], and in the excellent combinatorial strengthening [10], this proximity takes on an arithmetic nature encoded by the Gowers uniformity norms and related objects. Although we do not take the combinatorial viewpoint here, relying instead on a restriction estimate as in [25], it seems that combinatorial methods such as those of [10] would be necessary to extend our result to the case of longer progressions.

In Section 2.1 we set up notation and describe the approach.

2.1 Notation and approach

Throughout, µ will always refer to a (possibly complex) Radon measure supported on

[0, 1] ≈ T. We denote the variation norm of the measure µ by kµk.

19 ∗ For a set A and measure µ, set µA := µ|A the restriction of µ to the set A, and let Aµ denote the set of points of positive density with respect to the measure µ; that is, for each

∗ x ∈ Aµ, lim sup |µ|(A ∩ B(x, r))/|µ|(B(x, r)) > 0.

∗ Note that µ(A/Aµ) = 0 by [40], pg. 91, Remark 1.

Throughout, let (φn) be an approximate identity on T (that is, a sequence of positive functions with L1-norm 1 converging weak∗ to the dirac delta function), with the property

n n that supp(φcn) ⊂ [−2 , 2 ] and φcn|[−2n−1,2n−1] ≡ 1.

The primary additional technical aspect of the argument involves the “progression counting functional” Λ3(µ), deﬁned as Z lim φn ∗ µ(x)φn ∗ µ(x − r)φn ∗ µ(x − 2r) dx dr n → ∞

which ﬁrst appeared in the context of measures, in a slightly diﬀerent form, in [35], and more

generally the measure ∩3µ deﬁned by Z Z 3 f d ∩ µ := lim f(x, r)φn ∗ µ(x)φn ∗ µ(x − r)φn ∗ µ(x − 2r) dx dr. n → ∞

We develop their existence theory in Section 2.5.

Recall that the quantity α in (a) is a Hausdorﬀ dimension estimate, while the quantity

β of (b) is an estimate on the Fourier dimension of µ.

1 Theorem 2.1.1. Suppose that µ is a measure on T satisfying (a) and (b) with β > 2 and

9−2β α > 8 .

2 Then the ﬁnite measure ∩3µ : C(T ) → C given by

Z 3 g 7→ g(x, r) d ∩ µ(x, r) := Λg(µ) (2.1) Z := lim g(x, r)φn ∗ µ(x)φn ∗ µ(x − r)φn ∗ µ(x − 2r) dx dr n → ∞

20 is well-deﬁned.

1 Lemma 2.1.2. Suppose that the probability measure µ satisﬁes (a) and (b) for some β > 2

9−2β and α > 8 . Then the trivial progressions of step size 0 lie outside the support of the measure ∩3µ, or in other words

Z 3 1{0}(r) d ∩ µ = 0.

In Section 2.5 we show that the measure ∩3µ exists and that it gives no mass to the

degenerate progressions of step size zero; the main tools here are a Littlewood-Paley type

decomposition, the Fourier decay assumption b, and the use of uniformity norm-type esti-

mates to bound progression-counting multilinear functionals. In Section 2.2, we show that

3 the measure ∩ µA exists and that it is supported on the set of (x, r) for which x, x+r, x+2r is

∗ an arithmetic progression contained in Aµ. Section 2.4 contains the proof of Theorem 2.0.1. In Section 2.3, we obtain the proof of Lemma 2.3.2, which is the main ingredient in proving

the results of Section 2.2; the tools here are Bohr sets and the restriction estimate Theo-

rem 2.0.2.

2.2 Existence of the restricted measure on 3APs

In this section, for measurable sets A ⊂ T we demonstrate the existence of the measure

3 1 ∩ (µA) on T × T under the assumption that µ satisﬁes (a) and (b) with β > 2 and α suﬃ-

3 ciently close to 1, and we discuss the support properties of the measure ∩ (µA). Throughout

this section, we will be assuming the results of Section 2.5, speciﬁcally the truth of Theo-

rem 2.1.1 and Lemma 2.1.2.

1 Lemma 2.2.1. Suppose that µ is a probability measure on T which satisﬁes (b) for a β > 2

9−2β 3 and a for α > 8 , and that the set A ⊂ T is measurable. Then the Radon measure ∩ µA exists.

21 Proof. Consider the more general measure

3 ∗ d ∩ (µg)(x, r) := w − lim φn ∗ (g dµ)(x)φn ∗ (g dµ)(x − r)φn ∗ (g dµ)(x − 2r) dx dr

for a bounded measurable function g, provided the above weak∗ limit exists. First note that

∞ 3 we have boundedness of g ∈ L (µ) 7→ ∩ (µg) on its domain of deﬁnition, since for any

2 f ∈ C(T ) Z 3 | f d ∩ (µg)| Z = lim | f(x, r)φn ∗ (g dµ)(x)φn ∗ (g dµ)(x − r)φn ∗ (g dµ)(x − 2r) dx dr| n → ∞ Z 3 ≤ lim kfkL∞ kgkL∞ φn ∗ µ(x)φn ∗ µ(x − r)φn ∗ µ(x − 2r) dx dr n → ∞

3 3 =kfkL∞ kgkL∞ k ∩ µk

1 and for β > 2 and α close enough to 1 this last is finite by Theorem 2.1.1. Thus it suffices to show that the limit defining the operator

3 g 7→ ∩ (µg)

∞ L (µ) 7→M(T × T)

1 exists on a dense subset. Conditional on β > 2 and α being suﬀciently close to 1, by taking trigonometric polynomials as this dense subset we again use Theorem 2.1.1 to obtain

3 existence of ∩ (µg) for such g by applying linearity and the fact that for ξ ∈ Z, the measure dµ˜(x) := e2πiξx dµ(x) satisﬁes

α •| µ˜|(B(x, r)) . r

− β •| µb˜(ξ)| . (1 + |ξ|) 2 .

∞ 3 3 Thus for any g ∈ L , ∩ µg exists, and so in particular does ∩ µA.

3 We turn now to the support properties of the measure ∩ (µA).

22 Lemma 2.2.2. Suppose that µ is a probability measure on T satisfying (a) and (b) with

1 9−2β β > 2 and a value of α > 8 . Suppose further that the set A ⊂ T is measurable with µ(A) > 0. Then we have

3 ∗ ∩ µA(T × T \ {(x, r) ∈ T ×(T \{0}): x, x + r, x + 2r ∈ Aµ} = 0

∗ where we recall from Section 2.1 that Aµ denotes the points of positive density of the set A with respect to the measure µ.

Before proving Lemma 2.2.2, we need the following fact about the projection of ∩3µ to the x-coordinate.10 Interestingly, it seems inaccessible without recourse to restriction esimates

(a direct application of the U 2-techniques of Section 2.5 fails, for instance), leaving open the question of what shape, if any, an analogue would take in the case of longer progressions.

2 R 2 R 3 3 Let D µ denote the measure T f(x) dD µ(x) := f(x) d ∩ µ(x, r) provided that ∩ µ exists.

Proposition 2.2.3. Suppose that µ is a probability measure on T satisfying (a) and (b)

1 with β > 2 and a value of α > 1 − β/4. Suppose further that the set A ⊂ T is measurable

3 with µ(A) > 0, and that the measure ∩ µA exists. Then we have

2 D µA µA.

∞ R 2 R 2 1 Proof. For f ∈ L we will show that f dD µA . ( |f| dµA) 2 , which applied to indicator functions of sets implies absolute continuity.

By Dominated Convergence, it suﬃces to show the above for the dense subclass of f with fb ∈ `1(Z). Fix such an f.

10 k Q In the discrete case, it is known that the dual function D f := x 7→ E~u ι∈{0,1}k\{0} f(x − ι · u) of a K function f with L2 (m) bounds is in L∞(m)(m-denoting Haar measure), or indeed, that the dual function is continuous (see Lemma 2.6 of [31]). What we show may be thought of as a singular U 2 version of this fact.

23 Taking Fourier transforms,

2 Z Y X f(x) φn ∗ µA(x − ir) dr dx = φcn(ξ)φcn(2ξ)µcA(ξ)µcA(2ξ)fφ\n ∗ µA(ξ). (2.2) i=0 ξ∈Z We have by the triangle inequality that

φcn(ξ)φcn(2ξ)µcA(ξ)µcA(2ξ)fφ\n ∗ µA(ξ) ≤ µcA(ξ)µcA(2ξ) |fb| ∗ |µcA| (ξ) .

1 1 1 Choosing + + = 1 with pi > p◦ (when β > 0, this is possible by taking pi = 3 for p1 p2 p3

each i provided α > 1 − β/4 so that p◦ < 3)

X µcA(ξ)µcA(2ξ) |fb| ∗ |µcA| (ξ) ξ∈Z

≤kµcAk`p1 (Z)kµcAk`p2 (2 Z)k|fb| ∗ |µcA| k`p3 (Z)

1/2 1/2 3/2 . [µ(A)] [µ(A)] kfbk`1 kµcAk`p3 < [µ(A)] kfbk`1 < ∞

where in the ﬁrst inequality of the last line, we have used the restriction estimate Theorem

2.0.2 and Young’s convolution inequality, and in the second inequality we have applied

Theorem 2.0.2 again.

This shows absolute convergence of the sum in (2.2).

But then evaluating (2.2) in the limit as n → ∞, again applying the restriction estimate

Theorem 2.0.2 we get

Z 2 X f dD µA = µA(ξ)µA(2ξ)fµdA(ξ) c c ξ∈Z

≤kµcAk`p1 (Z)kµcAk`p2 (2 Z)kfµdAk`p3 (Z) Z 1/2 1/2 1/2 2 . [µ(A)] [µ(A)] |f| dµA ≤ kfkL2(µ), as was to be shown.

24 1 3 Proof of Lemma 2.2.2. If β > 2 and α is close enough to 1, then by Lemma 2.2.1 ∩ µA exists. We show ﬁrst that

3 ∗ c ∩ µA {(x, r) ∈ T × T : x, x + r, x + 2r ∈ Aµ} = 0.

Suppose that E ⊂ T × T is a collection of (x, r) which do not belong to the above set.

∗ Then for each (x, r) ∈ E, one of {x, x + r, x + 2r} ∈/ Aµ. We will decompose E into three sets corresponding to each of these three possibilities, E = E0 ∪ E00 ∪ E000.

0 ∗ We consider the case of E , the other cases being similar. Recall that µ(Aµ) = 0.

∗ 0 Let B = Aµ be the projection of E onto the x-axis.

3 c 2 c 2 c 3 c Then since ∩ µA(B ×T) = D µA(B ), D µA µA and µA(B ) = 0, we have ∩ µA(B ×

T) = 0. (The cases of E00 and of E000 require the analogous variants of Proposition 2.2.3, which we leave to the reader.)

3 ∗ Thus ∩ (µA) is in fact concentrated on the (x, r) such that x, x + r, x + 2r ∈ Aµ.

3 All that remains now is to show that ∩ (µA)(T ×{0}) = 0. But this is immediate from the obvious inequality

3 3 ∩ (µA) ≤ ∩ (µ)

and Lemma 2.1.2.

2.3 The Main Estimate : Proof of Lemma 2.3.2

Recall that for a given frequency set S ⊂ Z and radius 1 ≥ > 0, a Bohr set B = B(S, ) is deﬁned by

2πiξx B = {x ∈ T : |e − 1| < for all ξ ∈ S}

and (see, e.g., Section 4.4 of [49]) that

|S| . |B|. (2.3)

25 We will need the following bound.

Lemma 2.3.1. Suppose that the measure µ on T satisﬁes (a) and b. Then if S ⊂ Z is a

−1 ﬁnite set of frequencies and B = |B| 1B is the normalized indicator of the Bohr set B of radius and frequency set S, one has

−1 −(1−α)|S|(1−α+ β ) kB ∗ B ∗ µkL∞ . 2

Proof. Set µ1 = B ∗ B ∗ µ.

Let N ∈ N be a large integer to be speciﬁed in a moment.

2 2 2 2 Since kBb k`1 = kBbk`2 = kBkL2 < ∞ and |µb1| ≤ |Bb| , we may apply Fourier inversion to decompose

X 2 2πiξx X 2 2πiξx |µ1(x)| ≤ |B ∗ B ∗ µ(x)| ≤ | Bb(ξ) µb(ξ)e | + | Bb(ξ) µb(ξ)e | |ξ|≤N |ξ|>N =I + II.

Taking ψ a molliﬁer with ψ ≈ N1B(0,N −1), ψb ≈ 1B(0,N), we have

X 2πiξx I ≈ ψbB\∗ B ∗ µ(ξ)e = |ψ ∗ B ∗ B ∗ µ| ≤ ψ ∗ |µ|

1−α .N (2.4)

1−α where in the last inequality, we have used kψ ∗ |µ|kL∞ . N which follows from the deﬁnition of ψ and the Hausdorﬀ dimension assumption on the positive measure |µ| (for details see, e.g., Lemma 4.1 of [15]).

Further

X 2 X 2 − β II ≤ |Bb| |µb| ≤ ( |Bb| )kµbk`∞(B(0,N)c ) . N 2 kBkL2 |ξ|>N − β −|S| ≤N 2 (2.5)

26 by Plancherel together with (2.3).

Combining (2.4) and (2.5) we obtain

1−α − β −|S| kµ1kL∞ . N + N 2 .

This is minimized when both terms on the right are equal. Thus we let

β −1 log N = −|S| 1 − α + log , 2

at which point we ﬁnd

−1 −(1−α)|S|(1−α+ β ) kµ1kL∞ . 2 . (2.6)

Taking α ↑ 1 in the above estimate, we see that if β > 0, kµ1kL∞ . 1 for α suﬃciently

close to 1, and a closer inspection yields the bound lim supα → 1 kµ1kL∞ ≤ 2 in the limit as

max(CH ,CF ) ↓ 1.

Lemma 2.3.2. Suppose that the probability measure µ on T satisﬁes (a) and (b) and that

A ⊂ T satisﬁes

µ(A) > δ. (2.7)

1 Suppose further that β > 2 . Then there is a lower bound α0 on α ∈ (0, 1), depending on

β, δ, CH , and CF , and a number c = c(δ) > 0 such that if

α > α0 then

Λ3(µA) > c.

27 Proof. Let , η > 0. Let

S = {ξ ∈ Z : |1dAµ(ξ)| ≥ η} be the large spectrum of µA and let

B = {x : |e2πiξx − 1| < ∀ξ ∈ S} be the associated Bohr set of radius .

Set 1 B := 1 . |B| B

By the lower bound (2.3) on the size of a Bohr set

−|S| kBkL∞ . . (2.8)

We ﬁrst show that |S| can be bounded only in terms of η. This is a consequence of the restriction estimate Theorem 2.0.2. We have for p in the range guaranteed by Theorem 2.0.2 that

X X Z |S|ηp ≤ |µ (ξ)|p ≤ |1 µ(ξ)|p |1 µ(ξ)|p k1 kp ≤ 1 cA dA . dA . A L2(µ) ξ∈S where in the second to last inequality we applied the restriction estimate, and in the inequality before it we invoked Lemma 2.3.3 from the end of this section.

From the above we obtain that for appropriate p,

−p |S| . η . (2.9)

We will compare µA to B ∗ B ∗ µA =: µ1 and ﬁnd that they contain approximately the same number of 3-term progressions. By assuming α suﬃciently close to 1 in terms of δ in

3 2 order to bound kµ1kL∞ , a standard lower bound on ∩ (µ1)(T ) will then give us the result.

28 We next obtain a lower bound on Λ3(µ1). According to Lemma 2.3.1,

−1 −(1−α)|S|(1−α+ β ) kB ∗ B ∗ µAkL∞ ≤ kB ∗ B ∗ µkL∞ . 2 .

0 −1 So by choosing α ≥ α0 := β/(|S| ln( ) − 1), we ﬁnd that kµ1kL∞ ≤ U for some constant

U ↓ 2 as α ↑ 1,CH ,CF ↓ 1. We suppose that we have done so.

By a version of the dense Roth’s theorem on T due in the discrete case to Varnavides [54] (see also Proposition 2.2 of [22] for a general statement or Lemma 6.1 of [9] for a derivation of the continuous inequality from the discrete), we have

kµ1kL1 Λ3(µ1) ≥ c (2.10) kµ1kL∞ where c(x) increases as x increases. Using the bound on kµ1kL∞ from (2.6) and that kµ1kL1 >

δ since B is normalized, we see that c ≥ c(δ/U) > 0 remains bounded below as a function of δ > 0 as ↓ 0 provided α is chosen as above.

We would then be done if we knew that |Λ3(µA) − Λ3(µ1)| < c, so we now bound this quantity, keeping in mind that we are free to set the parameters , η to be anything we want provided α depends on them as speciﬁed.

By writing Λ3(µA) and Λ3(µ1) on the Fourier side, we have

X 2 4 2 |Λ3(µA) − Λ3(µ1)| = µcA(ξ) µcA(−2ξ)(1 − Bb(ξ) Bb(−2ξ) ) (2.11) ξ∈Z X 2 4 2 X 2 4 2 ≤ |µcA(ξ)| |µcA(−2ξ)||1 − Bb(ξ) Bb(−2ξ) | + |µcA(ξ)| |µcA(−2ξ)||1 − Bb(ξ) Bb(−2ξ) | ξ∈S ξ∈ /S 4 2 2−p p+1 k1 − B(ξ) B(−2ξ) k ∞ |S| + kµ k kµ k . b b L (ξ∈S) cA `∞(Sc) cA `p+1 for any p > 0.

∞ 2−p In the last line, we have used Holder’s inequality after pulling out the ` norm of |µcA| . We wish to choose p so that the restriction estimate of Theorem 2.0.2 may be applied to

29 bound kµcAk`p+1 by a constant, but we also want the exponent 2 − p to be positive. This is possible provided that

2 > p ≥ p◦ − 1

where p◦ is the critical exponent in Theorem 2.0.2.

Thus we want p◦ < 3. Observing the formula for p◦ given in Theorem 2.0.2, it suﬃces to

0 assume that α > 1 − β/4. So set α0 = max(α0, 1 − β/4) and assume α > α0. Choosing such a p in (2.11), and applying Theorem 2.0.2 and the bound

4 2 2 k1 − Bb(ξ) Bb(−2ξ) k`∞(ξ∈S) .

(cf. [25], Lemma 6.7) we obtain

2 2−p |Λ3(µA) − Λ3(µ1)| . |S| + η . (2.12)

By choosing and η appropriately, we can ﬁnish the proof by showing that |Λ3(µ1) −

Λ3(µA)| is smaller than the lower bound c of Λ3(µ1). To do this, we examine the estimate

(2.12).

Combining (2.9) with (2.12), we have

2 2−p 2 −p 2−p |Λ3(µA) − Λ3(µ1)| ≤ C |S| + Cη ≤ C η + Cη .

Since we have ﬁxed α depending on , η so that the bound in (2.10) is at least the constant c = c(δ/U) , we may choose ﬁrst η, and then small enough that the right hand side above

1 is less than, say, 2 c, and thus

1 1 Λ (µ ) > Λ (µ ) − c > c 3 A 3 1 2 2 giving us the result.

30 Remark 2.3.1. A computation shows that in the above theorem we may take η = =

(c/2C)1/(2−p), so that we can set

βηp ln(2)/2 p ln(2)/2 α0 = 1 − = 1 − β =: 1 − γ0(c)β. 0 ln(−1) − ηp ln(2) ln(−1) − p ln(2)

With CH ,CF close enough to 1 that we obtain a suﬃcient upper bound on kµ1kL∞ (and

thus lower bound on c) that c ∼ c(δ/2) we then have that for any ρ > 0 we may take

α − ρ ≥ max(1 − β/4, 1 − γ(δ)β) where γ(δ) = γ0(c(δ/2)). Since the constant C can be taken to be > 1, c ≤ 1, p < 2, and γ is growing as a function of p, we can calculate that

γ(1) ≤ 1/6 when µ(A) = 1, in which case Lemma 2.3.2 remains valid when α > 1 − β/6

(provided CH ,CF suﬃciently close to 1).

Lemma 2.3.3. Suppose that ν is a measure compactly supported in T d. Then for any p ∈ (1, ∞] X Z |ν(ξ)|p ≈ |ν(ξ)|p dξ. b d b d R ξ∈Z Proof. We ﬁrst prove that for p < ∞

Z X p p |ν| . |ν(ξ)| dξ b d b d R ξ∈Z (if p = ∞, the corresponding inequality is immediate).

Let g be any Schwartz function equal to 1 on the support of ν and for which supp gb ⊂ 1 1 d [− 2 , 2 ] . Then since dν = g dν, Z |ν(ξ)| = |ν ∗ g(ξ)| = | ν(ξ − η)g(η) dη| b b b 1 1 d b b [− 2 , 2 ] Z ≤kgkL∞ |ν(ξ − η)| dη. b 1 1 d b [− 2 , 2 ] R So |ν(ξ)| 1 1 d |ν(ξ − η)| dη. Thus by Holder’s inequality b . [− 2 , 2 ] b Z !p Z Z X p X X p p |ν(ξ)| . |ν(ξ − η)| dη ≤ |ν(ξ − η)| dη = |ν(ξ)| dξ. b 1 1 d b 1 1 d b d b d d [− , ] d [− , ] R ξ∈Z ξ∈Z 2 2 ξ∈Z 2 2

31 Now we prove the converse inequality. Fix a Schwartz function h equal to 1 on supp(ν)

d such that supp(h) ⊂ B(0, 1). Then for any ξ ∈ R

X ˆ νb(ξ) = hνc(ξ) = νb(n)h(ξ − n) d n∈Z by (111) of [55]. By a variant of Young’s inequality, we then have for any 1 ≤ p ≤ ∞ that ˆ kνbkLp ≤ kνbk`p khkL1 . Indeed, Z X ˆ ˆ kνbkL1 = νb(n)h(ξ − n) ≤ kνbk`1 khkL1 d n∈Z X ˆ ˆ kνkL∞ = sup ν(n)h(ξ − n) . kνk`∞ khkL1 b d b b ξ∈R d n∈Z P where we have used the ﬁrst part of the Lemma to bound d |hˆ(ξ − n)| by khˆk 1 , and n∈Z L interpolating between the two gives for any p ∈ [1, ∞]

kνbkLp . kνbk`p .

2.4 3APs in dense subsets : Proof of Theorem 2.0.1

The following is only a few lines removed from our main result, Theorem 2.0.1.

Proposition 2.4.1. Let δ > 0, and suppose that the probability measure µ on T satisﬁes

1 (b) with β > 2 and (a) with α suﬃciently close to 1 depending on β, δ, and the implicit constants CH and CF in (a) and b.

∗ Then if A is any measureable set with µ(A) > δ, Aµ contains 3APs; indeed, µ gives positive measure to the set of x such that x, x + r, x + 2r is a non-trivial 3AP contained

∗ within Aµ.

3 Proof. By Lemma 2.2.1, the measure ∩ (µA) exists and by Lemma 2.2.2 it is concentrated

∗ 3 inside a set identiﬁable with the non-trivial 3APs contained in Aµ. By Lemma 2.3.2, ∩ (µA)

32 ∗ is not the trivial measure and hence Aµ must contain 3APs. By Proposition 2.2.3, µ gives positive measure to the collection of base points x for such 3APs.

∗ Proof of Theorem 2.0.1. Let L = Aµ \ A denote the collection of points of density of A not contained in A. Of course, µ(L) = 0. We may thus ﬁnd an arbitrarily small open set around

0 0∗ L since µ is Radon; in particular, we may ﬁnd a subset A ⊂ A such that Aµ ⊂ A and such that

µ(A0) > δ

remains valid.

0∗ Applying Proposition 2.4.1, we ﬁnd that Aµ contains 3APs and that µ gives the collection of their basepoints positive measure. Consequently so too for A.

2.5 Existence of the measure ∩3µ : Proof of Theorem 2.1.1 and Lemma 2.1.2

The goal of this section is to prove Theorem 2.1.1 and Lemma 2.1.2.

We will need the following bound, which follows from the Hausdorﬀ dimension condition

(a) via straightforward arguments (see Lemma 4.1 of [15]).

(1−α)n kφn ∗ µk`∞ . 2 . (2.13)

2 For bounded functions f : T → R and g : T → R, deﬁne Z Λg(f) := g(x, r)f(x)f(x − r)f(x − 2r) dx dr

and for three such functions f1, f2, f3

Z Λg(f1, f2, f3) := g(x, r)f1(x)f2(x − r)f3(x − 2r) dx dr.

33 Following Gowers [21], set also

∆1f(x; r) = f(x)f(x − r)

∆2f(x; r, s) = ∆1f(x; r)∆1f(x − s; r) = f(x)f(x − r)f(x − s)f(x − r − s)

∆1g(x, y; r) = g(x, y)g(x, y − r)

∆2g(x, y; r, s) = ∆1g(x, y; r)∆1g(x − s, y − s; r)

=g(x, y)g(x, y − r)g(x − s, y − s)g(x − s, y − r − s).

One easily veriﬁes the following standard identity.

2 Lemma 2.5.1. Let f1 and f2 be bounded functions on T and g be bounded on T . Then

X Λg(f1) − Λg(f2) = Λg(h1, h2, h3) f1,f2 where the symbol P refers to a sum taken over the set f1,f2

{(f1 − f2, f1, f1), (f2, f1 − f2, f1), (f2, f2, f1 − f2)}.

We postpone the proofs of the following lemmas to the end of this section.

Lemma 2.5.2. Let f be a function on T which satisﬁes

ˆ − β |f(ξ)| ≤ CF (1 + |ξ|) 2 for all ξ ∈ Z (2.14)

supp(fˆ) ⊂ B(0, 2n+1) \ B(0, 2n) (2.15) for some β > 0.

Then for all η ∈ Z

−(2β−1)n |∆d2f(ξ; η)| . 2 and

supp ∆d2f(0; ·) ⊂ B(0, 2n+4).

34 2 Lemma 2.5.3. Let f, h1, h2 : T → R and g : T → R be bounded functions. Then Z

g(x, r)h1(x)h2(x − r)f(x − 2r) dx dr

Z Z 2 !1/4 1 1 ≤kh1kL∞ kh2kL∞ ∆ g(2x − r, x − r; −s)∆ f(r; 2s) dr dx ds .

Lemma 2.5.4. Let f be a function on T which satisﬁes

ˆ − β |f(ξ)| ≤ CF (1 + |ξ|) 2 for all ξ ∈ Z . (2.16)

supp(fˆ) ⊂ B(0, 2n+1) \ B(0, 2n) (2.17)

∞ 2 for some β > 0. Then for any h1, h2 ∈ L (T) and g : T → R, for all p ∈ [1, ∞]

− 1 (2β−1− 2 )n 1 4 p0 2 4 |Λg(h1, h2, f)| 2 kh1kL∞ kh2kL∞ k∆ g\(0, ξ; η)k 1 p . . `ξ`η

Lemma 2.5.5. Suppose that µ is a measure on T satisfying (b) with β > 0. Then for any

2 g : T → R and p ∈ [1, ∞], for all N > m ∈ N, N−1 ! − 1 (2β−1− 2 )n 1 X 4 p0 2 2 4 Λg(φN ∗ µ) − Λg(φm ∗ µ) . 2 max kφn+i ∗ µkL∞ k∆dg(0, ξ; η)k 1 p ι=0,1 `ξ`η n=m

1 P P p p where the mixed norm kG(ξ, η)k 1 p := |G(ξ, η)| . `ξ`η ξ η

Using the above lemmas we can complete the proof of Theorem 2.1.1 and Lemma 2.1.2.

Proof of Theorem 2.1.1. It suﬃces to show that the functional deﬁned by the limit in (2.1)

2 exists for g in a dense subclass of C(T ) and is bounded on that subclass. We take as our subclass the collection of all trigonometric polynomials, and by linearity it suﬃces to check

existence and boundedness for a monomial of the form g(x, r) = e2πi(xξ0+rη0).

To show existence of the limit, it suﬃces to show that given > 0 there is some m such

that for all large N,

Λg(φN ∗ µ) − Λg(φm ∗ µ) < . (2.18)

35 To this end, we apply Lemma 2.5.5 with p = 1. By our assumption on g, it is easy to see

that

2 k∆dgk`1 = 1.

Inserting this and the bound (2.13) into Lemma 2.5.5 gives

N−1 N−1 X 2(1−α)(n+1) −(2β−1)n/4 X −n( 2β−1 −2(1−α)) Λg(φN ∗ µ) − Λg(φm ∗ µ) . 2 2 ≈ 2 4 (2.19) n=m n=m

2β−1 1 9−2β and the exponent −( 4 − 2(1 − α)) will be negative when β > 2 and α > 8 . Since the sum (2.19) is a geometric series and remains ﬁnite in the limit N ↑ ∞, we can

take m large enough to guarantee that (2.18) holds, as desired.

Thus Z lim g(x, r)φn ∗ µ(x)φn ∗ µ(x − r)φn ∗ µ(x − 2r) dx dr n → ∞

exists for g a trigonometric polynomial. In particular, the limit exists for for g ≡ 1.

Finally, we complete the proof by showing that the above limit is bounded where deﬁned.

Indeed, setting

M := Λ1(|µ|)

we have

Z | lim g(x, r)φn ∗ µ(x)φn ∗ µ(x − r)φn ∗ µ(x − 2r) dx dr| n → ∞ Z ≤kgkL∞ lim φn ∗ |µ|(x)φn ∗ |µ|(x − r)φn ∗ |µ|(x − 2r) dx dr n → ∞

≤MkgkL∞

using that the limit deﬁning Λg(|µ|) exists for the trigonometric polynomial g ≡ 1.

1 1 Proof of Lemma 2.1.2. Choose a compactly supported Schwartz function g with [− 2 , 2 ] ⊂

−1 supp g ⊂ [−1, 1]. Let > 0 and set g(r) = g((4 ) r).

It is not hard to see that ∆kg is rapidly decaying since g is Schwartz.

36 We have

3 ∩ µ({|r| ≤ }) ≤ |Λg (µ)|. (2.20)

We will show that

p−1 2 p Λg (µ) .

for some p > 1 to be speciﬁed later.

R 3 Since Λg(φn ∗ µ) → g d ∩ µ by Theorem 2.1.1, we can send N → ∞ and set m = 0 in

the bound from Lemma 2.5.5, which gives

 2β−1− 2 !  ∞ p0 Z − 4 −2(1−α) n 1 3 X 2 4 | g d ∩ µ − Λg (φ0 ∗ µ)| .  2  k∆[g(0; ·)k`p n=0

1 9−2β for some p > 1. Since β > 2 , provided that α > 8 we may ﬁnd p > 1 such that the sum in the right hand side above is ﬁnite. Fix such a p. We then have

Z 1 1 3 2 4 2 4 g d ∩ µ . Λg (φ0 ∗ µ) + k∆[g(0; ·)k`p = C +k∆[g(0; ·)k`p

2 We are now done, since for any function G on [0, 1]2 with Gˆ ∈ L1(R ), for p > 1 if

−1 G(~x) = G((4 ) ~x) then applying Lemma 2.3.3 and a change of variables,

2 p−1 ˆ ˆ p ˆ kGk`p . kGkLp = kGkLp

and G = ∆2g(0; ·) is such a function.

All that remains is to prove Lemmas 2.5.2, 2.5.3, 2.5.4, and 2.5.5.

Proof of Lemma 2.5.2. The (trivial) identity

1 ˆ ˆ ∆df(η1; τ) = f(τ)f(η1 + τ)

together with the Fourier decay condition (2.14) and the support condition (2.15) give

−β/2 −β/2 1 n n+1 |∆df(η1; τ)| . 1{2 ≤|τ|,|τ+η1|≤2 } (1 + |τ|) (1 + |η1 + τ|) (2.21)

37 One readily veriﬁes that

2 X 1 1 ∆df(ξ; η) = ∆df(−η1; τ)∆df(−ξ − η1; τ − η2) τ∈Z which shows with (2.21) that η ∈ B(0, 2n+4), and it follows by Cauchy-Schwarz that

Applying (2.21) to the above gives

1 ! 2 X −β −β 2 n n+1 |∆df(ξ; η)| . 1{2 ≤|τ|,|τ+η1|≤2 } (1 + |τ|) (1 + |η1 + τ|) τ∈Z 1 ! 2 X −β −β n n+1 + 1{2 ≤|τ|,|τ+ξ+η1|≤2 } (1 + |τ|) (1 + |ξ + η1 + τ|) τ∈Z −(2β−1)n .2 .

Proof of Lemma 2.5.3. We have

g(x, r)h1(x)h2(x − r)f(x − 2r) dx dr

1 Z Z 2 2 ≤kh1kL∞ | g(x, r)h2(x − r)f(x − 2r) dr| dx

1 Z 2 =kh1kL∞ g(x, r)g(x, s + r)h2(x − r)h2(x − r − s)f(x − 2r)f(x − 2r − 2s) dr dx ds

1 Z 2 1 1 1 =kh1kL∞ 4 g(x, r; −s)4 h2(x − r; s)4 f(x − 2r; 2s) dr dx ds

Z Z 2 !1/4 1 1 ≤kh1kL∞ kh2kL∞ 4 g(x + r, r; −s)4 f(x − r; 2s) dr dx ds

Z Z 2 !1/4 1 1 =kh1kL∞ kh2kL∞ ∆ g(2x − r, x − r; −s)∆ f(r; 2s) dr dx ds

38 where in the last line we have applied the change of variables r 7→ −r + x.

Proof of Lemma 2.5.4. By Lemma 2.5.3,

g(x, r)h1(x)h2(x − r)f(x − 2r) dx dr

Z Z 2 !1/4 1 1 ≤kh1kL∞ kh2kL∞ ∆ g(2x − r, x − r; −s)∆ f(r; 2s) dr dx ds

Z 1/4 2 2 =kh1kL∞ kh2kL∞ ∆ g(2x − r, x − r; −s)∆ f(r; 2s, t) dx dr ds dt .

Applying Plancherel gives

ZZ 2 1 1 2 ∆ g(x + r, x; s, t)∆ f(r; s, t) dr dt ds dx 2 2

X 2 2 =2 ∆dg(0, 2ξ; η1/2, η2)∆df(ξ; η) 2 ξ∈Z,η∈Z 1   p0 2 X 2 p0 k∆ g(0, ξ; η1/2, η2)k 1 p sup |∆ f(ξ; η)| (2.22) . d `ξ`η  d  ξ∈Z 2 η∈Z by Holder’s inequality.

By Lemma 2.5.2, this is bounded by

2 2 −(2β−1− 0 )n k∆ g(0, ξ; η1/2, η2)k 1 p 2 p d `ξ`η and two applications of Lemma 2.3.3 in order to move to the continuous context, change variables, and back, give the result.

39 Proof of Lemma 2.5.5. We decompose into telescoping series and invoke Lemma 2.5.1

N−1 X Λg(φN ∗ µ) − Λg(φm ∗ µ) = Λg(φn+1 ∗ µ) − Λg(φn ∗ µ) n=m N−1 X X = Λg(h1, h2, h3) φ ∗µ,φ ∗µ n=m n+1 n N−1 X X ≤ Λg(h1, h2, h3) . (2.23) φ ∗µ,φ ∗µ n=m n+1 n

We next calculate a bound on each individual term in the above sum. By a linear change

11 of variables, it suﬃces to suppose that h3 = φn+1 ∗ µ − φn ∗ µ := µn.

We can apply Lemma 2.5.4, which gives

2β−1− 2 p0 1 − 4 n 2 4 |Λg(h1, h2, µn)| 2 kh1kL∞ kh2kL∞ k∆dg(0, ξ; η)k 1 p . . `ξ`η

Inserting this into the right hand side of (2.23) gives the result.

2.6 Further Remarks

It is interesting to note that if the dependence of the Hausdorﬀ dimension α in Theo- rem 2.0.1 on the measure δ of the set A were dropped, we would in fact have that µ-almost every point x was the starter of a three-term arithmetic progression contained in the support of µ. It is not a priori clear whether this dependence is necessary, or whether for some α,

β strictly less than 1, and some CH , CF , it might be true that whenever µ satisﬁed (a) and b that every A for which µ(A) > 0 must contain 3APs. In the discrete context, Green’s result applies to any dense subset of the primes, regardless of density, but the primes have

“dimension” 1 within the natural numbers. On the other hand Conlon, Fox, and Zhao [10]

11The change of variables may result in a different g. However, since the change of variable is linear, it will affect the bounds we obtain by at most a constant which doesn’t affect the conclusion of the theorem.

40 have developed a combinatorial approach strong enough to obtain a relative Szemer´edithe-

orem (kAPs for k ≥ 3) for subsets of pseudorandom sets of integers having “dimension”

less than 1 within the integers, and their result applies for any dense subset of a suﬃciently

pseudorandom set, which gives reasonably strong evidence that such a result may be possible

in our context.

There is some further weak evidence the dependence of α on δ may be unnecessary, in

the form of a result of Conlon and Gowers [22] (see also [34] for the k = 3 case) that as

N ↑ ∞, with probability converging to 1, every random subset E of [N] large enough for the statement to not trivially fail has the property that all size δ|E| subsets of E contain kAPs. The natural cutoﬀ for 3APs in this result of Conlon and Gowers is that the set

1 E ⊂ [N] satisfy |E| ≈ N 2 . Thus it is natural to conjecture that the appropriate bound on

1 the Hausdorﬀ dimension α of a measure µ on T in order that it contain 3APs be α > 2 .

1 Though we have achieved results for any β > 2 , we require a far larger α. In a diﬀerent direction, it is natural to ask whether the collection of r for which ther exist x such that x, x + r, x + 2r lies in supp(µ), or indeed in A for a set A of positive µ-measure, be large in Hausdorﬀ dimension. We obtain results along these lines in Chapter 3.

In [9], we applied a notion of “higher order” Fourier dimension to extend the results of

[35] to longer progressions. It would be natural to extend the results of the present thesis to this setting, namely, to demonstrate that given a measure of (k−1)st order Fourier dimension

suﬃciently close to 1, that any subset of suﬃciently large measure must contain kAPs. The

obvious approach to such a result would be to adapt the methods of [10].

41 Chapter 3: On the number of 3APs in fractal subsets of Euclidean Space

This chapter is primarily work joint with Steven Senger. √ In the discrete setting, a set E ⊂ [N] := [0,...,N − 1] of ≈ N 1−1/ log N elements12

need not contain non-trival three term arithmetic progressions, as shown by Behrend in [2].

However if E contains ≈ N 1−(log log N)4/(log N)2 elements then it must contain (non-trivial)

3APs (Bloom 2016, [5]). If on the other hand the set possesses some pseudorandomness, such as enough Fourier decay that

4 4 X 1 3 k1E − δEk 2 = Eb(ξ) ≤ δ U 2 E ξ6=0∈ZN

then much smaller densities, δE, suﬃce for the set to contain non-trivial 3APs - such a set

− 1 E contains non-trivial 3APs once δE is greater than N 2 (this follows from Gowers’ proof,

[21]).

In this chapter we explore applications of techniques developed in relation to the Falconer

Distance Conjecture (a question on the distribution of the two-point conﬁgurations within a

set) to questions of 3AP counts in both the ﬁnite and fractal settings. We obtain largeness

results for the set of lengths of step-sizes of 3APs occurring within certain sets and obtain a

generalization (Theorem 3.0.8) ofLabaand Pramanik’s result in [35] with an assumption on

12 Given numbers A and B depending on some parameter or set of parameters t, we use the notation A . B denote the existence of an unspeciﬁed constant C, independent of t, for which the inequality A ≤ CB holds. We use the notation A ≈ B to denote the simultaneous inequalities A . B and B . A.

42 q the L norm of µb in place of a Fourier decay assumption, which relies heavily on techniques and tools from the discrete world (primarily Bohr sets and the Arithmetic Regularity Lemma

of Green and Tao ([27])).

3.0.1 Integer sets

The integer version of a question of Croot asks

Question 3.0.1 (Croot, question 7.6 of [12]). Given C, D > 0, is it true that for all suﬃ-

ciently large N a subset E of [N] satisfying

|E| 1 ≥ N logC N

and

Eb ≤ logD N `1 must contain a non-trivial three-term arithmetic progression?

We cannot answer this diﬃcult question. Instead, we study the easier question of how much pseudorandomness as measured by the `q norm of Eb is necessary to guarantee 3APs, as q varies in [2, 3]. We prove

|E| 1 − 2 Theorem 3.0.1. Let E ⊂ [N] be a set with density δ = N N which satisﬁes a bound

Eb ≤ δM `q where ! T − 1 C q ≤ q(M, δ) := 2 + min 3 1 − , 1 T >1 T (2) δ T (2) δ T C2c M ln C2c M (see Deﬁnition 3.0.4 for the deﬁnition of the L2 Varnavides function c(2).) Then provided N

is suﬃciently large depending on q and M the set E contains non-trivial 3APs.

43 Alternatively, for any q ∈ [2, 3] and δ > 0 there is an M = M(δ, q) ∈ [0, ∞] such that

for all suﬃciently large N a set E ⊂ [N] of density ≥ δ with Eb < M contains a non- `q √ trivial 3AP, and M is monotonic decreasing in its second argument with M(δ, 3) = 3 2δ and

M(δ, 2) = ∞.

3.0.2 Fractal sets

At a rough level, an analog of Hausdorﬀ dimension for subsets of integers E ⊂ [N] is

the smallest α for which |E| ≤ N α. Then rephrasing the discussion at the start of the

present chapter, informally Behrend’s result shows that even a Hausdorﬀ dimension of 1 is

insuﬃcient to gaurantee 3APs, but the corollary of Gowers’ proof shows that good enough

Fourier decay does yield results.

The above phrasing strongly suggests the question: What about in the fractal setting?

In 2008 Keleti constructed a full dimensional subset of [0, 1] containing no 3APs ([33]). In

2009,Labaand Pramanik ([35]) proved that a probability measure µ on [0, 1] satisfying

α µ(B(x, r)) ≤ CH r (3.1)

−β/2 |µb(ξ)| ≤ CF |ξ| (3.2)

for β > 2/3 and α suﬃciently large depending on α and β must contain 3APs in its support.

In 2016, Carnovale showed that under these conditions, µ must give positive measure to

the set of starting points for 3APs in its support ([8]). This is the content of Chapter 2.

Also in 2016,Laba, Pramanik, and Henriot ([30]) showed, amongst other things, the 2009

Laba-Pramanik result holds when β is taken to be any positive number. Following these

results, we conjecture the following.

d Conjecture 3.0.2. Suppose that the compactly supported probability measure µ on R sat- isﬁes conditions (3.1) and (3.2) and that α > 2d/3. Then under appropriate quantitative

44 assumptions on the parameters involved, the set of r for which there exists an x and a u with

|u| = r such that x, x + u, x + 2u is a 3AP contained within supp µ has positive Lebesgue measure.

Deﬁnition 3.0.3. Let X be a compact abelian group with Haar measure m. We deﬁne the

∞ 3 multilinear functional Λ3 :(L (G)) → C via the formula ZZ Λ3(f0, f1, f2) = f0(x)f1(x − r)f2(x − 2r) dm(x) dm(r).

Given a single f ∈ L∞(G), we deﬁne

Λ3(f) = Λ3(f, f, f).

Deﬁnition 3.0.4. The Varnavides function c : (0, ∞) →(0, ∞) is given by

n o c(t) = inf Λ (f): f ≥ 0, kfk = 1, kfk ≤ t . 3 L1(T) L∞(T)

The L2 Varnavides function is the function c(2) : (0, ∞) →(0, ∞) given by

n o c(2)(t) = inf Λ (f): f ≥ 0, kfk ≥ t, kfk = 1 . 3 L1(T) L2(T)

The following is the celebrated Varnavides’ Theorem, from [54].

Theorem 3.0.5. [Varnavides’ Theorem.] Let 0 < δ < 1. Then there exists a constant,

N = N(δ) > 0 such that for any subset E ⊂ {0, 1,...,N}d, of size |E| > δN d, there are

2d & N three-term arithmetic progressions in E.

A discretization argument together with Theorem 3.0.5 gives that the Varnavides function c above does indeed map (0, ∞) to (0, ∞), while the same statement for the function c(2) will be given as Corollary 3.1.1 in Section 3.1. We summarize the equivalent functional form of the ﬁrst inequality below.

45 d d Corollary 3.0.6. Let X ∈ T , ZN . Then for any M, δ > 0 there exists a constant

c = c(M, δ) > 0 with the following property. Let f : X →[0,M] satisfy kfkL1(X) ≥ δ. Then

Λ3(f) ≥ c.

Throughout, given an element r ∈ T, we will write

|r| = |r| = min (r mod 1, (1 − r) mod 1) , T

d and similarly for r = (r1, . . . , rd) ∈ T we deﬁne

1 d ! 2 X 2 |r| = |r| d = |r | . T i T i=1

To state our main result, we deﬁne relaxations of conditions (3.1) and (3.2) suited for

our purposes:

(a) Iα(|µ|) < ∞

− β d (b) ||cµ|(ξ)| ≤ CF (1 + |ξ|) 2 for all ξ ∈ Z ,

where we remind the reader of the Riesz potential,

ZZ −α X 2 −(d−α) Iα(µ) := |x − y| dµ(x)dµ(y) ≈ |µb(ξ)| |ξ| . d ξ∈Z We need to deﬁne the following notation for the set of r > 0 for which there exists at

least one 3AP of diﬀerence with length r in a given set,

43(E) := {r > 0 : x, x + u, x + 2u ∈ E, |u| = r}.

Note that studying the size of 43(supp µ) as a notion of “largeness” for the parameter

set

{r : there exists x such that x, x + r, x + 2r ∈ supp µ}

46 seems natural in light of Falconer’s Distance Conjecture, which conjectures that the

42(supp µ) := {r : there exists x and a u with |u| = r such that x, x + u ∈ supp(µ)} have positive Lebesgue measure whenever the Hausdorﬀ dimension, α, of µ is greater than d/2 and d ≥ 2. A sharpness result of Falconer ([16]) shows that dimension d/2 is best possible, and Mattila ([39]) showed that a weaker Fourier decay condition than in (b) guarantees the truth of the conclusion of the Falconer Distance Conjecture for a measure µ when α + β > d

(an approach which has been central to progress since). As necessarily the set of step-sizes of 3APs within supp µ is contained within the set of diﬀerences between points in supp µ, optimal size estimates on the former under a given set of hypotheses is strictly harder than for the latter.

d Theorem 3.0.7. Let µ be a Radon probability measure on T satisfying (a) and (b). Then if

d − 1 > (4d − 3β)/2,

(2β + d − α)/β < 3,

47 13 and α is suﬃciently close to d depending on the quantities CF , β, and Iα(µ), the bilinear

distance set 43(supp µ) has positive Lebesgue measure.

2 q We also prove the following L -type bound under an assumption that µb ∈ ` for some q ∈ [2, 3].

Theorem 3.0.8. Suppose that µ is a positive measure on [0, 1]d with

(d) kµbkLq ≤ M

for some δ > 0, M < ∞ and q ∈ [2, 3]. Then there is a number q0 = q0(M, δ) ∈ (2, 3] with ! T − 1 C1 q0(M, δ) = 2 + min 3 1 − , T >1 T (2) δ T (2) δ T C2c M ln C2c M (2) 2 for some constants C1 and C2 and where c is the L Varnavides function, such that if q ≤ q0 the set

3 ∆ (supp µ) = {r ∈ R : x, x + u, x + 2u ∈ supp µ, |u| = r} has positive s-dimensional Hausdorﬀ measure for all s ≤ 1 satisfying

1 d(6 − 2q) s < + , 2 2q

13Throughout this chapter we will often use phrases of the form “α is suﬃciently close to d depending on the parameters [β, C, X, Y , etc.]” in situations where satisfyingly explicit bounds are unavailable. In the case at hand, for instance, what can be said is the following. In Theorem 3.0.7, provided that 2 |µb(ξ)| α ≥ α0,Iα(µ) ≤ Iα0 , sup β ≤ CF d ξ∈Z |ξ| for some bounds 0 < α0,Iα0 < ∞, and p N(d−α0) c0 := c Iα0 CF 2

where c is the Varnavides function, it suﬃces that 12C I F α0 −(β0−(d−α0))N/2 2 < c0 1 − 2−(β0−2(d−α0))/2 for some N ∈ N. We will see in the proof that by increasing α0 towards d such an N may always be found provided that β0 > 2(d − α0).

48 and positive Lebesgue measure when d ≥ 2.

Further, when d ≥ 2 the natural measure on this set derived from µ (see the deﬁnition

of δ (µ) below) possesses a weighted L2 density with respect to the one-dimensional Lebesgue

measure.

Laba-Pramanik type results like Theorem 3.0.7 follow directly from Theorem 3.0.8, as

the following corollary makes precise. This is one indication of the naturality of the `q-norm

condition on µb.

Corollary 3.0.9. Suppose that the positive measure µ satisﬁes (a) and (b) where α is

suﬃciently close to d depending on the parameters Iα(µ), CF , and β. Then supp µ contains

non-trivial 3APs, and indeed the set

3 ∆ (supp µ) = {r ∈ R : x, x + u, x + 2u ∈ supp µ, |u| = r}

has positive Lebesgue measure if d ≥ 2 and positive s-dimensional Hausdorﬀ measure for any

1 d(2β+4α−4d) s ≤ 1 satisfying s < 2 + 4(β+d−α) .

β+d−α Proof. By Lemma 3.2.3, we have that for q1 = 2 β

q1 q1−2 −(q1−2)β/2 d−α q1−2 kµk q ≤ I (µ)C sup |ξ| |ξ| = I (µ)C , b ` 1 α F α F ξ

β d−α p β+d−α β+d−α kµk q ≤ I (µ) C . (3.3) b ` 1 α F

Consider the function q0(M) := q(M, 1) of Theorem 3.0.8. If we set

M = M(Iα(µ),CF , α, β)

to be the right-hand-side of (3.3), then we see that as Iα(µ) stays bounded above by a p parameter CH , and as d − α ↓ 0, M ↓ Iα(µ) and so q0(M) nears some q0 ∈ (2, 3).

49 β+d−α Meanwhile, q1 = 2 β ↓ 2, so if d − α is small enough, q1 ≤ q0 and the conclusion of Theorem 3.0.8 holds for the measure µ.

3.0.3 Finite ﬁelds

A very similar strategy to that of Section 3.2, following Iosevich and Rudnev’s adaptation,

[32], of Mattila’s paper [39], yields analogs of Theorem 3.0.7 in ﬁnite ﬁelds. This setting has

the advantage of exposing a clear dependence of the bounds in these theorems on the total

number of 3APs contained in the set under question (as made precise by Corollary 3.3.16

and Lemma 3.3.10), a feature obscured in the continuous setting.

d Let Fq denote the d-dimensional vector space over the ﬁnite ﬁeld with q elements, and

d for a subset E ⊂ Fq , let

d 3AP (E) =: (x, r) ∈ Fq : x, x + r, x + 2r ∈ E .

Many expectations regarding 3APs in ﬁnite ﬁelds were recently shattered by the following

result ([11], [13]).

Theorem 3.0.10. [Croot-Lev-Pach-Ellenberg-Gijswijt] There is a γ ∈ (0, 1) such that if d

d γd is suﬃciently large and E ⊂ Fq with |E| ≥ q then E contains a non-trivial 3AP.

Building on this work, Fox and Lovasz have recently proved some powerful results in [17],

which we discuss further below. One consequence of their work is the following corollary,

which gives us some quantitative bounds on the number of 3APs present in a suﬃciently

d large subset of Fq .

d αd Corollary 3.0.11. There is a cq > 0 such that if E ⊂ Fq with |E| > q then for d suﬃciently large

|3AP (E)| ≥ q−cq(1−α)dq2d.

50 In line with this, we have the following bound on the number of distances arising from

the step-sizes in 3AP (E).

d αd Theorem 3.0.12. Let E ⊂ Fq with |E| ≥ q . Then if c α > q 1 + cq

where cq is the constant from Corollary 3.0.11 and d is suﬃciently large

3 4 (E) & q.

In a private communication, Hans Parshall pointed out that we cannot hope to replace

& with = in Theorem 3.0.12 using size conditions alone. This is a consequence Lemma 17

d in [38], which shows that there exist dense subsets of Fq not containining a given isometric copy of a ﬁxed“non-spherical” conﬁguration.

It is well-known that if the set E is Fourier-pseudorandom, then E contains the “ex-

pected” number of three-term arithmetic progressions. Following Iosevich and Rudnev in

[32], we deﬁne the following class of sets:

d Deﬁnition 3.0.13. Say that E is a Salem set if for all non-zero m ∈ Fq

−d 1 2 Eb(m) . q |E| .

d d/2 Theorem 3.0.14. If E ⊂ Fq is a Salem set and |E| & q then

3 4 (E) & q.

3.1 Integer results

d 2 A non-zero positive measure µ on T with µb ∈ ` must necessarily contain within its support non-trivial three-term arithmetic progressions: the qualitative statement follows from

51 the observation that since µ ∈ L2, supp µ is a set of positive Lebesgue measure, together with the statement that such sets contain non-trivial 3APs (and indeed aﬃne images of any

ﬁnite point conﬁguration), a consequence of the Lebesgue density theorem and the pigeon- hole principle. The quantitative statement is the following consequence of the arithmetic regularity lemma, whose discussion we postpone until Subsection 3.1.1.

d d Corollary 3.1.1. Suppose that X = T or ZN , and that f : X →[0, ∞) satisﬁes kfkL1 ≥

(2) δ, kfkL2 < M < ∞. Then there exists a number c = c (M, δ) such that

Λ3(f) ≥ c.

3 On the other hand, if µb ∈ ` with suﬃciently small norm, then again lower bounds on

Λ3(µ) are available.

d d Lemma 3.1.2. Suppose that µ is a positive measure on T or ZN with mass kµk ≥ δ and √ 3 kµbk`3 ≤ 1 + cδ. Then 3 Λ3(µ) ≥ (1 − c)δ .

Proof. We have

√ 3 3 3 3 X 3 3 X 3 1 + cδ ≥ kµbk`3 = µb(0) + |µb(ξ)| ≥ δ + |µb(ξ)| ξ6=0 ξ6=0 so √ 3 kµbk`3(ξ6=0) ≤ cδ.

Further, note by Holder that !2/3 !1/3 X 2 X 3 X 3 3 µb(ξ) µb(−2ξ) ≤ |µb(ξ)| |µb(2ξ)| ≤ kµbk`3(ξ6=0) . ξ6=0 ξ6=0 ξ6=0 Then

3 X 2 3 X 2 3 Λ3(µ) − δ = µb(ξ) µb(−2ξ) − δ = µb(ξ) µb(−2ξ) ≤ cδ . ξ ξ6=0

52 d In summary: If µ is a positive, non-zero measure on T and kµbk`p = M, then if M is

suﬃciently small and p = 3 then Λ3(µ) > 0, while if p = 2 then no bound whatsoever on

M is required to conclude that Λ3(µ) > 0. It is natural to ask whether one can interpolate

between these results for p ∈ (2, 3). Indeed we can, as Proposition 3.1.3 below shows.

Proposition 3.1.3 (Mass lemma from `ˆp bounds). Let δ, M > 0. Then there exists a

q = q(M, δ) and a number c = c(q)(M, δ) > 0 with ! T − 1 C q(M, δ) = 2 + min 3 1 − , 1 T >1 T (2) δ T (2) δ T C2c M ln C2c M (2) δ 2 for some constants C1,C2 and where c ( M ) is the L Varnavides function, such that the following holds. Suppose that µ is a positive measure on [0, 1]d or [N]d with kµk ≥ δ and kµbk`q ≤ M. Then (q) Λ3(µ) ≥ c (M, δ).

Equivalently, for any q ∈ [2, 3] there is an M = M(q) and a c = c(q)(M, δ) > 0 such that

if kµk ≥ δ and kµbk`q ≤ M then Λ3(µ) ≥ c, with M(q) ↑ ∞ as q ↓ 2.

Theorem 3.0.1 follows as a corollary.

Proof of Theorem 3.0.1. Set µ = 1E . Then kµk ≤ M and kµk = 1, so by Proposition 3.1.3, δ b `q

(q) Λ3(µ) ≥ c (M, 1)

independent of N.

On the other hand, the trivial 3APs within E contribute

2 1 X Y 1 |E| N µ(x)µ(x)µ(x) = = = δ−2N −1 = o (1) N 2 N 2 δ3 |E|2 N x∈[N] i=0

− 1 since δ N 2 .

For large enough N, this will be less than c(q)(M, 1), guaranteeing the existence of a non-trivial 3AP within E.

53 The proof of Proposition 3.1.3 will use the following lemma, whose proof we delay until

the end of the present section.

Lemma 3.1.4. For all ∈ (0, 1) and C > 1 there exists a q = q(, C) ∈ (2, 3) such that for

d all borel measures f with fˆ ∈ `q(Z ) there exists a decomposition

f = g + h such that

• g ≥ 0,

•k gkL1 = kfk,

ˆ ˆ • gˆ , h . f ,

ˆ • gˆ `2 . C f `q ,

and

ˆ • h `3 . .

Further, if we specify an upper bound M for the `q norm of fˆ, we may take ! T − 1 2 ln C q = 2 + min 3 1 − , . (3.4) T >1 T M T M T ln Proof of Proposition 3.1.3. Set f = µ.

Fix a number > 0 depending on M and δ to be speciﬁed later.

We use Lemma 3.1.4 with C ≈ 4 to ﬁnd a q = q(M, ) ∈ (2, 3) and a decomposition

f = g + h

54 where ! T − 1 C0 q = 2 + min 3 1 − , (3.5) T >1 T M T M T ln g ≥ 0

ˆ ˆ |gˆ| , h . f ,

ˆ kgˆk`2 ≤ 4 f ≤ 4M, `q

ˆ ˆ kgˆk`3 ≤ f ≤ f ≤ 4M, `3 `q ˆ h . . (3.6) `3

Rescaling g by (4M)−1, Corollary 3.1.1 gives

δ Λ (g) ≥ (4M)3 c(2) . (3.7) 3 4M

We have

X ˆ ˆ ˆ ˆ ˆ ˆ |Λ3(f) − Λ3(g)| . |Λ3(f0, f1, f − g)| ≤ f0(ξ) h(ξ) f1(−2ξ) ≤ h f0 f1 `3 `3 `3 where f0, f1 ∈ {f, g} are chosen to maximize the estimate on the right-hand side of the ﬁrst inequality. Thus applying (3.6)

2 |Λ3(f) − Λ3(g)| . M whence using (3.7) we have

3 (2) 2 Λ3(f) ≥ 4M c (δ/4M) − O M .

So choosing

= C004Mc(2) (δ/4M) for suﬃciently small constant C0 gives the result with

(q) 3 (2) c (M, δ) & M c (δ/4M) .

55 1 Proof of Lemma 3.1.4. Fix λ ∈ (0, kfkL1 ) and a number η ∈ (0, 2 ) to be speciﬁed later. Let q ∈ (2, 3) be a number to be speciﬁed later, and suppose that

fˆ ≤ M. `q

Set

Eλ = {|f| ≥ λ} .

It would be natural to letg ˆ be the restriction of fˆ to the set where it is large, but it seems unlikely that the resulting g would be a positive function. Instead, a variant of this

approach replacing 1Eλ by the Fourier transform of the corresponding Bohr set works. Let

2πix·ξ B = B (Eλ, η) = x : e − 1 ≤ η∀ξ ∈ Eλ be the Bohr set with frequency set Eλ and radus η, and let

φ = 1B/|B|.

We recall the following facts about Bohr sets

|E | (i) |B| & η λ (see, e.g, Lemma 4.4 in [49])

p−1 −|Eλ| −|Eλ| p and consequently kφkL∞ . η , and more generally kφkLp . η , which com- bined with the Hausdorﬀ-Young inequality gives

−|E | 1 φb . η λ p `p

for p ≥ 2.

2 (ii) 1 − φb(ξ) . η for all ξ ∈ Eλ (see, e.g., the proof of Lemma 6.7 in [25]).

Set

g = φ ∗ f,

56 h = f − g.

We have q −q ˆ |Eλ| ≤ λ f `q so that by (i) for p ≥ 2

q q −λ−qkfˆk 1 −( M ) 1 . φb . η `q p ≤ η λ p (3.8) `p

This gives by Holder’s inequality

1/2 1/2 2 2 2 q X − M q−2 ˆ ˆ ˆ ( λ ) 2 kgˆk`2 = φb f f ≤ f φb 2 . Mη . (3.9) `q ` q−2

Meanwhile by (ii), the triangle inequality, and that φb ≤ 1

3 X 3 3 hˆ = 1 − φb(ξ) fˆ(ξ) `3 ξ 3 3 3 3 X ˆ X ˆ = 1 − φb(ξ) f(ξ) + 1 − φb(ξ) f(ξ) c Eλ Eλ 3 3 6 X ˆ X ˆ .η f(ξ) + f(ξ) c Eλ Eλ 3 X q ≤η6 fˆ + λ3−q fˆ(ξ) q ` c Eλ

c ˆ where in the last line we have used that q < 3 and that on Eλ, f ≤ λ. Thus

q 6 3−q 3−q ˆ ˆ 3 2 q/3 h . η 3 M + λ 3 f ≤ η M + λ 3 M . (3.10) `3 `q

Setting

λ = Mη

(3.9) and (3.10) become

δ q q−2 (3−q)q q−2 −(( λ ) 2 ) −(η 2 ) kgˆk`2 . η M = η M (3.11)

3−q 3−q ˆ 2 3 q/3 h . η M + (ηM) M . η 3 M, (3.12) `3

57 where in (3.12) we have used the bounds η < 1 and q ≥ 1.

Using these bounds we now must check that g and h fulﬁll the requirements of this lemma

for some choice of q ∈ (2, 3). Since g = φ ∗ f, obviously g ≥ 0 and kgkL1 = kfkL1 , and

ˆ ˆ h . f , so we need to check that for an appropriate choice of constant η ∈ (0, 1) and q ∈ (2, 3) the desired norm bounds are attained.

Solving

(3.12) ≤

gives 3 3−q η ≤ M So we may take T η = M

3 T −1 for a choice of T ≥ 3−q , or q ≤ 3 1 − T . Solving

(3.11) ≤ CM

for q yields that we must (and that it is suﬃcient to) take ! T − 1 2 ln C T − 1 2 ln C q ≤ 2 + min 3 1 − , = 2 + min 3 1 − , , T η−1 ln η−1 T −T −T M ln M where the minimum is taken over all T , showing that we may always take q > 2. This

completes the proof.

3.1.1 3AP counts of L2 functions

Deﬁne |f(x) − f(y)| kfkLip = kfkL∞ + sup x6=y |x − y| and Z Z 2 !1/4

kfk 2 = f(x)f(x − r) dx dr . U

58 d d Given numbers M and N, say that θ ∈ T is (M,N)-irrational if for all q ∈ Z with kqk ≤ M, kq · θk ≥ M . `1 T N An analysis of the proof of Theorem 1.2 of [27] reveals that only L2-bounds on the function f are needed, and further that the same proof works over [N]d, so that a slight modiﬁcation yields the following.

2 d Lemma 3.1.5 (L Regularity Lemma in [N] ). Suppose that f :[N] →[0, ∞) with kfkL1 ≥

2 δ > 0, f ∈ L with kfkL2 ≤ 1. Let F : N → N be an increasing function and let > 0. Then there exists an M ∈ N and a decomposition f = fstr + fU 2 + fL2 such that

kfU 2 kU 2 ≤ F(M),

kfL2 kL2 ≤ , and

fstr(n) = F (n/N, n mod q, θn) where

d F : [0, 1] × Z /q Z × T → [0, 1],

d q, d, kF kLip ≤ M, and θ ∈ T is (F(M),N)-irrational.

We ﬁnd it convenient to use Herglotz’s Theorem in the proof of Corollary 3.1.1.

Definition 3.1.6. A continuous function ψ on a locally compact abelian group G is positive- definite if for all finite complex-valued sequences an

X ψ(i − j)aiaj ≥ 0. i,j Theorem 3.1.7. [Herglotz’s Theorem] Let f be a positive finite Borel measure on a locally compact abelian group. Then fˆ is positive-definite on the Pontryagin dual Gˆ of G. Con- verseley, if ψ is a positive-definite function on a locally compact abelian group, then ψ is the

Fourier transform of a positive ﬁnite Borel measure on the dual group.

59 Proof of Corollary 3.1.1. We deal with the case that X = [N]d ﬁrst. Let > 0. Use the

Regularity Lemma to write f = fstr + fU 2 + fL2 , where

kfU 2 kU 2 ≤ F(M),

kfL2 kL2 ≤ ,

and

fstr(n) = F (n/N, n mod q, θn)

where

d F : [0, 1] × Z /q Z × T → [0, 1],

d q, d, kF kLip ≤ M, and θ ∈ T is (F(M),N)-irrational. Then by following exactly the proof in [27] of their Theorem 6.1 (Szemeredi’s Theorem),

replacing their use of the Regularity Lemma for 1-bounded functions by Lemma 3.1.5, one

obtains the claim.

d Now if instead X = [0, 1]d, we embed [0, 1]d ⊂ [−1, 2]d ,→ T , noting that this embedding preserves the 3AP count (abusing notation, we continue to write f for the resulting function

d deﬁned on T ). Let > 0. Let N be such that

1   2

X ˆ  f(ξ)  ≤ d ξ∈Z ,|ξ|>N/100

and write fN = φN ∗ f were φ is a molliﬁer satisfying φcN |B(0,N/2) ≡ 1, supp φcN ⊂ B(0,N), so that kf − fN kL2 ≤ . We have that

3 |Λ3(f) − Λ3(fN )| .

60 since

X ˆ 2 ˆ ˆ d ˆ |Λ3(f) − Λ3(fN )| = f(ξ) f(−2ξ) − fN (ξ) fN (−2ξ) d ξ∈Z

X ˆ 2 ˆ ˆ d ˆ X ˆ 2 ˆ ˆ d ˆ ≤ f(ξ) f(−2ξ) − fN (ξ) fN (−2ξ) + f(ξ) f(−2ξ) − fN (ξ) fN (−2ξ)

|ξ|≤N/2 |ξ|>N/2

3 3 X ˆ 2 ˆ ˆ ˆ 3 ≤0 + 2 f(ξ) f(−2ξ) . f ≤ f ≤ . `3(|ξ|>N/2) `2(|ξ|>N/2) |ξ|>N/2

−1 d Let p ∈ (10N, 20N) be a prime and g denote the function on (p Zp) given by

X ˆ 2πiξ·x g(x) = fN (ξ)e . d ξ∈B(0,N)⊂(Z /p Z)

−1 d d If we embed (p Zp) within T in the natural way, g(x) = fN (x); this identiﬁcation may be sidestepped, however - note that

ˆ 1.ˆg is positive-deﬁnite since by Herglotz’s Theorem fN is,

2. thus g ≥ 0 again by Herglotz’s Theorem,

ˆ 3. kgkL1 =g ˆ(0) = fN (0) = kfkL1 ,

4. kgkL2 = kfN kL2 ≤ kfkL2 ≤ 1 by Plancherel’s theorem,

5.Λ 3(g) = Λ3(fN ) by the identity

X 2 Λ3(g) = gˆ(ξ) gˆ(−2ξ) ξ ˆ applied ﬁrst to g, then to fN , and using thatg ˆ = fN under the identiﬁcation Zp =

p−1 p−1 − 2 ,..., 0,..., 2 ⊂ Z.

By the conclusion of this lemma over [p]d, we therefore obtain that

3 (2) Λ3(f) + O( ) = Λ3(fN ) = Λ3(g) ≥ c (kfkL1 ) .

61 3.2 Fractal results

d Let µ be a Radon measure supported in T or [0, 1]d. Whenever this integral exists, deﬁne

ZZ σ (µ)(ρ) := µb(2η)µb(η − ρθ)µb(η + ρθ) dσ(θ) dη, where the σ(θ) inside the integral is the Lebesgue measure on the unit sphere. Also, deﬁne

ς (µ)(ρ) = µb(2η)µb(η − ρθ)µb(η + ρθ) dσ(θ) dη, and ZZ Λ3(f) := f(x)f(x − r)f(x − 2r)dxdr, when these integrals exist.

In what follows, we will freely use (φn)n∈N as an approximate identity with supp φcn ⊂

B(0, 2n). Finally, let δ (µ) be the measure on R deﬁned as

Z Z Y g(r) dδ (µ)(r) := lim g(|u|) φn ∗ µ(x − iu) dx du n → ∞ i∈[3] provided that the right hand side above deﬁnes a continuous linear functional on C(R) (as

d it does, e.g., if µ has a density function f ∈ C(R )).

d We note that the measures on [0, 1]d may be identiﬁed with (a subset of) those on T via

d the embedding given by ﬁrst mapping [0, 1]d to the unit cube in R , followed by identifying

d d the cube [−2, 2] with T in the natural way. This identiﬁcation preserves the energies Iα(µ)

p and the L norms kµbkLp up to multiplicative constants, and also preserves the 3AP counts as measured by Λ3(µ) and the (up to rescaling) set of lengths of 3APs as measured by δ (µ)

d when the deﬁnition of δ (µ) is extended to T in the obvious way. In this section we will perform this identiﬁcation without further comment.

62 3.2.1 Proof of Theorem 3.0.7

The proof of Theorem 3.0.7 follows from Lemma 3.2.1 (Support), Lemma 3.2.6 (Mass), and Lemma 3.2.10 (Non-singularity), below.

Support Lemma

We state the following without proof.

d Lemma 3.2.1. [Support lemma.] Let µ be a probability measure on T and suppose that δ (µ) exists and has ﬁnite total mass. Then

supp(δ (µ)) ⊂ {r > 0 : x, x + u, x + 2u ∈ supp(µ), |u| = r} .

Mass Lemma

In order to prove the Mass Lemma, we need the following technical results.

d Lemma 3.2.2. Let µ be a probability measure on [0, 1] satisfying Iα(µ) < ∞, and (φn)n∈N

n an approximate identity with supp φcn ∈ B(0, 2 ). Then for all n

p n(d−α)/2 kφn ∗ µkL∞ ≤ Iα(µ)2 .

1 Proof. Since φn ∗ µ has an ` Fourier transform, we have by the triangle inequality and

Cauchy-Schwarz

X 2πiξ·x |φn ∗ µ(x)| = φcn(ξ)µb(ξ)e

X −(d−α)/2 (d−α)/2 ≤ |µb(ξ)| |ξ| |ξ| |ξ|≤2n p n(d−α)/2 ≤ Iα(µ)2 .

d Lemma 3.2.3. Suppose that the measure µ on T satisﬁes (b) for β > 0. Then

q q−2 d−α kµbk`q ≤ Iα(µ) sup |µb(ξ)| |ξ| . ξ

63 In particular, µ ∈ `q for q = 2 + 2 d−α = 2 β+d−α . b β β

Proof. We have

X q |µb| d ξ∈Z X 2 −(d−α) q−2 d−α = |µb| |ξ| |µb| |ξ| d ξ∈Z q−2 d−α ≤Iα(µ) sup |µb(ξ)| |ξ| ξ

From here forward, we deﬁne J := J(d−2)/2, the Bessel function of order (d − 2)/2. We

record some well-known Bessel function estimates in the following lemma, which may be

found in [39] (2.1),(2.2), and the ﬁrst displayed formula on page 216.

Lemma 3.2.4. Consider the Bessel function J = J(d−2)/2.

1. For all ξ ∈ R

−1/2 Jb(ξ) ≤ |ξ| .

2. For all ξ ∈ R

(d−2)/2 Jb(ξ) ≤ |ξ| .

d 3. There is a constant C so that the following holds. Let g ∈ C(R) and G ∈ C(R ) satisfy G(u) = g(|u|). Then

Z ∞ −(d−2)/2 Gb(ξ) = C |ξ| rd/2J(r |ξ|)g(r) dr. 0

d Lemma 3.2.5. For any f, g ∈ C(R ) with gˆ ∈ L1 and supp(g) ⊂ [0, ∞). Z Z Z g(r) dδ (f)(r) = O(1) g(r)rd/2 ρd/2J(rρ)σ (f)(ρ) dρ dr.

Further, if fb ∈ L1, Z δ (f)(r) = O(1)rd/2 ρd/2J(rρ)σ (f)(ρ) dρ.

64 Proof. Let G(u) = g(|u|). Then using Lemma 3.2.4

Z Z Y g dδ (f) = G(u) f(x − ir) dx dr i∈[3] Z = Gb(ξ)fˆ(η + ξ)fˆ(η − ξ)fˆ(2η) dη dξ Z Z ∞ =O(1) |ξ|−(d−2)/2 rd/2J(r |ξ|)g(r) dr fˆ(η + ξ)fˆ(η − ξ)fˆ(2η) dη dξ 0 Z ∞ Z =O(1) g(r) |r|d/2 |ξ|−(d−2)/2 J(r |ξ|)fˆ(η + ξ)fˆ(η − ξ)fˆ(2η) dη dξ dr 0 Z ∞ ZZZ =O(1) g(r) |r|d/2 sd−1s−(d−2)/2J(rs)fˆ(η + sθ)fˆ(η − sθ)fˆ(2η) dη dσ(θ) ds dr 0 Z ∞ Z =O(1) g(r) |r|d/2 sd/2J(rs)σ (f)(s) ds dr. 0

Under the hypothesis that fb ∈ L1, it is immediate that σ (f) is bounded, and thus that the

above gives the desired pointwise estimate.

Lemma 3.2.6. [Mass lemma.] Let µ be a probability measure satisfying (a) and (b) and suppose that the measure δ (µ) exists. Then if

(2β + d − α)/β < 3

and α is suﬃciently close to d depending on the quantities CF , β, and Iα(µ), the measure

δ (µ) has positive mass.

Proof. Let (φn) be an approximate identity satisfying the hypotheses of Lemma 3.2.2, so that for any n ∈ N

p n(d−α) kφn ∗ µkL∞ ≤ Iα(µ)2

and kφn ∗ µkL1 = 1.

Then by Varnavides’ Theorem there is a monotonic decreasing c = c (kφn ∗ µkL∞ ) ≥

p n(d−α) c( Iα(µ)2 ) (so that c is in particular independent of any other properties of the mea-

sure µ), such that

Λ3(φn ∗ µ) ≥ c.

65 Now ﬁx N ∈ N. Consider the diﬀerence

|kδ (µ)k − Λ3(φN ∗ µ)| = lim |Λ3(φn ∗ µ) − Λ3(φN ∗ µ)| n → ∞ X ≤ |Λ3(φn+1 ∗ µ) − Λ3(φn ∗ µ)| n≥N

Let

Fn = {(f, f, f − g) , (f, f − g, g) , (f − g, g, g)}

where f = φn+1 ∗ µ and g = φn ∗ µ, so that each triple in Fn has exactly one entry equal to

(φ − φ ) ∗ µ. Using the identity Λ (φ ∗ µ) − Λ (φ ∗ µ) = P Λ (f , f , f ), n+1 n 3 n+1 3 n (f0,f1,f2)∈Fn 3 0 1 2 Q Q P Q Q which follows from the identity ai − bi = j ij bj applied to the integrands, we bound

|Λ3(φn+1 ∗ µ) − Λ3(φn ∗ µ)|

≤12 sup |Λ3(f0, f1, f2)| . f~∈Fn

Taking the Fourier transform, applying Holder’s inequality and using the support properties of φdn+1 − φcn µb (which is equal to one of the fi), we have

X |Λ3(f0, f1, f2)| = fb0(η)fb1(η)fb2(2η) d η∈Z 2 2 1 Y ˆ Y 1/3 d−α 3 ≤ fi ≤ Iα(µ) µ(ξ) |ξ| `3(B(0,2n+1)\B(0,2n−1)) b `∞(|ξ|≈2n) i=0 i=0 where in the last line we have used Lemma 3.2.3. Together with (b) this gives

− β−2(d−α) n |Λ3(f0, f1, f2)| ≤ CF Iα(µ)2 2 .

This yields that

−(β−(d−α))N/2 −(β−2(d−α))/2 |kδ (µ)k − Λ3(φN ∗ µ)| ≤ 12CF Iα(µ)2 / 1 − 2 . (3.13)

66 Fix a lower bound α0 for α and an upper bound I0 for Iα(µ) such that the constant √ n(d−α0) c0 := c( I02 ) coming from Varnavides Theorem is positive. Fix also a lower bound

β0 for β. We may suppose that α0 and β0 satisfy the condition (2β0 + d − α0)/β0 < 3.

Let N be large enough that the expression

−(β0−(d−α0))N/2 −(β0−2(d−α0))/2 12CF I02 / 1 − 2 , which upper-bounds the right-hand-side of (3.13) is strictly less than c0, which is possible since β0 > 2(d − α0). Choose α > α0 suﬃciently close to d that the Varnavides bound c satisﬁes

p n(d−α) c( Iα(µ)2 ) ≥ c0, which is possible since c is decreasing in its argument. Putting these together, we obtain from the reverse triangle inequality that

kδ (µ)k = |Λ3(φN ∗ µ) − (Λ3(φN ∗ µ) − kδ (µ)k)|

≥Λ3(φN ∗ µ) − |kδ (µ)k − Λ3(φN ∗ µ)| > c0 − c0 = 0 showing that kδ (µ)k > 0.

β Remark 3.2.1. From the proof we see that we may take α > d − 2 provided that for some

N ∈ N and c = c(·) the lower bound function coming from Varnavides Theorem 3.0.5 the strict inequality 12C I p N(d−α0) F α0 −(β0−(d−α0))N/2 c Iα CF 2 > 2 0 (1 − 2−(β0−2(d−α0))/2) is satisﬁed for the bounds

α ≥ α0,

Iα(µ) ≤ Iα0 , 2 |µb(ξ)| sup β ≤ CF . d ξ∈Z |ξ| 67 Non-singularity Lemma

We ﬁrst establish existence of the measure δ (µ).

d Lemma 3.2.7. [Existence lemma.] Let µ be a Radon probability measure on T . Suppose

q that µb ∈ ` for some q ≤ 3. Then the measure δ (µ) exists. d+β−α In particular, if µ satisﬁes (a) and (b) then this holds when q = 2 β provided this value is less than or equal to 3.

Proof. Let (φn) be an approximate identity. That δ (µ) exists follows from: the absolute

convergence of the sums 2 X φ\n ∗ µ(ξ) φ\n ∗ µ(−2ξ) d ξ∈Z and more generally for any ﬁxed η0, η1 of

X φ\n ∗ µ(ξ − η0) φ\n ∗ µ(ξ − η1) φ\n ∗ µ(−2ξ) , d ξ∈Z the observation that this convergence guarantees the existence of the limits

2 ZZ Y lim f(x, r) φn ∗ µ(x − ir) dx dr (3.14) n → ∞ i=0

for any trigonometric monomial f(x, r) = e2πi(x·(η0−η1)+r·η1), hence for any trigonometric

polynomial f, hence for any continuous function f, and hence the existence of a measure

d d d ∩3µ on T × T represented by the action on C(T ) given by (3.14), and ﬁnally by the

observation that for a function g on R,

Z ∞ Z g(r) δ (µ)(r) = g(|u|) d ∩3 µ(x, u). 0

The following technical estimates will be used in proving the Non-singularity lemma.

68 Lemma 3.2.8. If ρ > d(1 − 1/q) then

Z d−1−ρ 3 r ς (µ)(r) dr . kµbkLq .

In particular, if µ sastisﬁes (b) and ρ > (4d − 3β)/2 then

Z d−1−ρ r ς (µ)(r) dr . 1.

1 Proof. Suppose ﬁrst that µb ∈ L . Converting from polar coordinates we have Z rd−1−ρς (µ)(r) dr ZZ −ρ ≤ |ξ| µb(ξ + η)µb(ξ − η)µb(η) dξ dη.

Using the Brascamp-Lieb inequality we have that this is bounded above by

≤ |ξ|−ρ kµk3 . Lp b Lq

The ﬁrst statement of the result then follows upon taking ρ > d(1−1/q) = d/q0 and applying

a limiting argument to the measures φn ∗ µ. Using the Fourier decay of µ, if 1/p + 3/q = 2,

q > 2d/β, and p > d/ρ this is ﬁnite precisely when ρ > (4d − 3β)/2, proving the second

statement.

d Proposition 3.2.9. Suppose f ∈ C(R ) is a positive function with fˆ ∈ L1. Let ρ < d − 1. Then Z ∞ d−1−ρ d−1−ρ δ (f)(s) . s r |σ (f)(r)| dr. 0

Proof. By Lemma 3.2.5 we have

Z ∞ δ (f)(r) = O(1)rd/2 sd/2J(rs)σ (f)(s) ds. 0

Split this integral as

Z 1/r Z ∞! d/2 d/2 δ (f)(r) . r + s J(rs) |σ (f)(s)| ds =: δ (f)1 (r) + δ (f)2 (r). 0 1/r

69 Then using Lemma 3.2.4

δ (f)1 (r) Z 1/r d/2 (d−2)/2 d/2 .r (sr) r |σ (f)(s)| ds 0 Z 1/r =rd−1 sd−1 |σ (f)(s)| ds 0 Z 1/r =rd−1 sρsd−1−ρ |σ (f)(s)| ds 0 Z 1/r ≤rd−1−ρ sd−1−ρ |σ (f)(s)| ds. 0

Similarly,

δ (f)2 (r) Z ∞ d/2 −1/2 d/2 .r (sr) r |σ (f)(s)| ds 1/r Z ∞ = (sr)(d−1)/2 |σ (f)(s)| ds 1/r Z ∞ = r(d−1)/2sρ−(d−1)/2sd−1−ρ |σ (f)(s)| ds 1/r Z ∞ ≤ rd−1sρsd−1−ρ |σ (f)(s)| ds 1/r Z ∞ ≤rd−1−ρ (rs)ρsd−1−ρ |σ (f)(s)| ds 1/r Z ∞ ≤rd−1−ρ sd−1−ρ |σ (f)(s)| ds. 1/r

d Lemma 3.2.10. [Non-singularity lemma.] Let µ be a Radon probability measure on R .

q Suppose that either µ satisﬁes (b) or that µb ∈ L . Let d−1 > (4d−3β)/2 or d−1 > d(1−1/q), respectively. Then the measure δ (µ) exists, and further δ (µ) ∈ L∞.

Proof. Existence follows from Lemma 3.2.7.

Let fn = φn ∗ µ. Let ρ satisfy either d − 1 > ρ > (4d − 3β)/2 or d − 1 > ρ > d(1 − 1/q)

as appropriate.

70 By Lemma 3.2.8 and the hypotheses on µ and ρ,

Z ∞ d−1−ρ r |σ (fn)(r)| dr < ∞ 0 uniformly in n, so by Lemma 3.2.9

sup δ (fn)(s) < ∞. n,s

Noting also that for functions g and f withg, ˆ fˆ ∈ L1

2 Z ZZ Y g(s) δ (f)(s) = g(|r|) f(x − ir) dx dr, i=0 we have by the deﬁnition of the measure δ (µ) that for any g ∈ L1(R) withg ˆ ∈ L1

2 Z ∞ Z Y g δ (µ) = lim g(|r|) fn(x − ir) dx dr n → ∞ 0 i=0 Z Z = lim g(s)δ (fn)(s) ds |g(s)| ds = kgk 1 n → ∞ . L and so δ (µ) ∈ L∞.

We also include here the following weaker result, valid when d = 1.

d q Lemma 3.2.11. Let µ be a Radon probability measure on [0, 1] . Suppose that µb ∈ L for some q ≤ 3. Then the measure δ (µ) exists, and further

s 3 H 4 (µ) & kδ (µ)k for any s ≤ 1 satisfying 1 d(6 − 2q) s < + . 2 2q

d+β−α In particular, if µ satisﬁes (a) and (b) then this holds with q = 2 β .

Proof. As in Lemma 3.2.10, existence follows from Lemma 3.2.7.

71 d Given a, R ∈ R and an > 0, let χ = 1B(a,R+) − 1B(a,R) be the indicator function of the annulus about the point a with inner radius R and outer radius R + . Then according to

Falconer’s Lemma 2.1 in [16] for any s ∈ (0, 1)

d−1 d−1 − d−1 −1 d−1 d−1 − d−1 s −(1−s) |χb(ξ)| . min R ,R 2 |ξ| 2 min(, |ξ| ) . min R ,R 2 |ξ| 2 |ξ| . (3.15)

By Plancherel’s theorem applied to the integral of the product of the functions F0(x, r) :=

χ(r)φn ∗ µ(x),F1(x, r) := φn ∗ µ(x − r)φn ∗ µ(x − 2r) Z Z χ δ (µ) = lim F0(x, r)F1(x, r) dx dr n → ∞ ZZ = lim χ(ξ − 2η)φ\n ∗ µ(ξ + η)φ\n ∗ µ(ξ)φ\n ∗ µ(η) dξ dη. n → ∞ b

By the Brascamp-Lieb inequality,

Z 3 3 χ δ (µ) ≤ lim kχkLp φ\n ∗ µ = kχkLp kµkLq n → ∞ b Lq b b

where 1 3 + = 2. p q

q 1 d(6−2q) So let p = 2q−3 . If we choose s < 2 + 2q then by (3.15) we have

d−1 s 2 kχbkLp . R .

This gives Z s d−1 δ (µ) ([R,R + ]) = χ δ (µ) . R 2 , and the Falconer type argument follows:

Suppose that ([R ,R + ]) is a covering of supp δ (µ) contained in B(0, 2). Then i i i i∈N

d X X s 0 < kδ (µ)k = δ (µ) [0, 1] ≤ δ (µ) ([Ri,Ri + i]) . i , i i showing that the s-dimensional Hausdorﬀ measure of supp(δ (µ)) is positive.

72 3.2.2 Proof of Theorem 3.0.8 Pointwise decay

d Lemma 3.2.12. Let σr denote the uniform measure on the unit sphere of radius r in R .

d Suppose that F : R → R and that F is compactly supported as a distribution. Let D =

supp Fb. Then Z Z −(d−1) |F (x)| dσr(x) .D r |F (x)| dx. ||x|−r|

Proof. Let φ be compactly supported in B(0,OD(1)) and such that φb|D = 1. Then F = φ∗F , so

Z Z Z |F | dσr = |φ ∗ F | dσr ≤ |φ| ∗ |F | dσr Z Z = |φ| ∗ σr |F | ≤ k|φ| ∗ σrkL∞ |F (x)| dx ||x|−r|

Lemma 3.2.13. Suppose that q < 3. Then

q−1 −(3d q −1) 3 ς (µ)(r) . min 1, r kµbkLq .

Proof. One estimates via Holder’s inequality

ς (µ)(r) = µb(η + rθ)µb(η − rθ)µb(2η) dη dσ(θ) Z 3 3 3 . kµbkL3 dσ(θ) = kµbkL3 . kµbkLq .

d Now ﬁx η ∈ R and let Fη(ξ) := µb(η + ξ)µb(η − ξ). Then supp(Fcη) ⊂ supp(µ) + supp(µ)

so that Fcη is compactly supported. By Lemma 3.2.12,

Z Z −(d−1) |Fη(ξ)| dσr(ξ) . r |Fη(ξ)| dξ. ||ξ|−r|

73 Thus ZZ ς (µ)(r) = |Fη(θ)| |µb(2η)| dσ(θ) dη ZZ −(d−1) .r 1B(r,O(1))(ξ) µb(η + ξ)µb(η − ξ)µb(2η) dξ dη

−(d−1) d/p 3 ≤r r kµbkLq for 1/p + 3/q = 2, where in the last line we have used the Brascamp-Lieb inequality. By hypothesis on q, this yields the desired inequality.

Tools from Mattila’s Book [41]

The following can be read from Mattila’s paper [39], though our reference will be Mattila’s

book [41].

In this section, suppose that

s ∈ L1 is non-negative and deﬁne Z d(r) := sd/2J(rs)s(s) ds

D(r) := r−(d−1)/2d(r)

S(s) := s(d−1)/2s(s) Z ∞ s−1 Is := s(r)r dr. 0

Since s ≥ 0, Is is non-decreasing in s.

The following is an immediate consequence of the discussion in Section 15.2 of [41] (where

s ≡ σ(µ), S ≡ Σ(µ), d ≡ δ(µ), and D ≡ ∆(µ)).

Lemma 3.2.14. Given the above,

√ Z ∞ √ D(r) = r sJ(rs)S(s) ds. 0 74 Further, there exist functions S,L, and K such that a). D = S + L

b). kSkL2 ≈ kSkL2

√ R ∞ √ c). L(r) = r 0 sS(s)K(rs) ds

−1/2 −3/2 d). |K(r)| . min(r , r ).

The L2 bound

We recall the notation of Subsubsection 3.2.2. Given a measure µ, set

Z Z s(r) := µ(2η)µ(η − rθ)µ(η + rθ) dσ(θ) dη, d b b b R Sd−1 Z ∞ d(r) := sd/2J(rs)s(s) ds 0 D(r) := r−(d−1)/2d(r)

S(s) := s(d−1)/2s(s) Z ∞ s−1 Is := s(r)r dr. 0

Lemma 3.2.15.

Z ∞ 2 2 2 L(r) dr . I(d−1)/2 + inf Os(1)Is : s > d/2 . 0

Proof. The proof (in the case of s ≡ σ(µ), S ≡ Σ(µ), d ≡ δ(µ), D ≡ ∆(µ), and Is ≡ Is(µ)) is

contained in the proof of Proposition 15.2.a) in [41]. Using Lemma 3.2.14 and the deﬁnitions

of the terms involved, the proof here is identical. In particular, note that Theorem 15.6 of

[41] goes through in the setting of this section.

−t Proposition 3.2.16. If Is < ∞ for s > d/2 and s(r) . r where s + t > d then

D ∈ L2.

75 Proof. The proof here is the same as in Mattila’s L2 Falconer result [39].

One has D = S+L and that kSkL2 ≈ kSkL2 by Lemma 3.2.14. Then using the hypothesis on s Z Z 2 2 d−1 2 d−1 −t kSkL2 ≈ kSkL2 = r s(r) dr . r s(r)r dr = Id−t ≤ Is < ∞

since d − t < s and I(·) is non-decreasing.

By Lemma 3.2.15 and the monotonicity of I(·),

2 2 2 2 kLkL2 . I(d−1)/2 + inf Os(1)Is : s > d/2 . Is .

This completes the proof.

d q Corollary 3.2.17. Suppose that µ is a probability measure on [0, 1] and that µb ∈ L for q ∈ (2, 3). Then D (µ) ∈ L2.

Proof. Let ρ ∈ (d(1 − 1/q), d) ⊂ (d/2, d). By Lemma 3.2.8,

Z d−1−ρ 3 Is (µ) (=) r ς (µ)(r) dr . kµbkLq

for s = d − ρ.

By Lemma 3.2.13

t s (µ)(r) . r

q−1 3−1 for t = 3d q − 1 ≥ 3d 3 − 1 = 3d − 1 ≥ d when d > 1. Since t > d we have that

s + t > d,

so by Lemma 3.2.16, D (µ) ∈ L2.

Proof of Theorem 3.0.8. By Proposition 3.2.17 D (µ)(r) = r−(d−1)/2δ (µ)(r) ∈ L2 which

shows that δ (µ) has no point masses. Informally this means δ (µ) concentrates no mass on

76 the diagonal (x, x, x) ∈ supp(µ)3. Similarly Lemma 3.2.1 states that δ (µ) is concentrated

on those r for which there exists some x, u, |u| = r for which (x, x + u, x + 2u) ∈ supp(µ)3.

Together, this shows that δ (µ) is concentrated on the lengths of step-sizes of the non-trivial

3APs contained in the support of µ.

(q) By Lemma 3.1.3, we have Λ3(µ) ≥ c (M, δ) > 0, where we may take ! T − 1 C1 q ≤ q0 = 2 + min 3 1 − , . T >1 T (2) δ T (2) δ T C2c M ln C2c M

Since kδ (µ)k = Λ3(µ), this gives that the mass of δ (µ) is not zero, which together with the

statement about the set on which δ (µ) is concentrated give the ﬁrst part of the theorem.

Finally, the statement about the s-dimensional Hausdorﬀ measure of supp(δ (µ)) is

Lemma 3.2.11. Together, these statements complete the proof of Theorem 3.0.8.

3.3 Finite ﬁeld results

3.3.1 Proof of Corollary 3.0.11

In [17], Fox and Lovasz prove the following quantitative triangle removal lemma. First,

a deﬁnition.

d Deﬁnition 3.3.1. Given X1,X2,X3 ⊂ Fq , a triple (x, y, z) ∈ X1 × X2 × X3 is a triangle if

x + y + z = 0.

d d Say that (x, y, z) is a triangle in Fq if it is a triangle in X1 × X2 × X3 where each Xi = Fq .

Theorem 3.3.2. [Theorem 3 of [17]] Suppose we have m = pn disjoint triangles, say

m n Cp m m m {(xi, yi, zi)}i=1, in Fp . Let δ = . Then, for X1 = {xi}i=1, X2 = {yi}i=1, and X3 = {zi}i=1,

2n we have at least δp triangles in X1 × X2 × X3.

Proof of Corollary 3.0.11. Per usual, we take X1 = E, X2 = E, and X3 = −2E so that a

triangle in X1 × X2 × X3 corresponds to a 3AP in E.

77 Considering the trivial 3APs in E, there are always at least |E| > qαd = q−(1−α)dqd

disjoint triangles in X1 × X2 × X3. By Lemma 3.3.2, there are then at least

q−cq(1−α)dq2d

triangles in X1 × X2 × X3, and correspondingly at least that many 3APs within E.

3.3.2 Gauss sums

By essentially the same proof given for Lemma 4.2 of [24] one has

Lemma 3.3.3. Suppose that E is Salem. Then

−3d 3 2d − d |3AP (E)| = q |E| q + O q 2 .

−2d 1 Proof. In the language of [24], E being Salem means that E is q |E| 2 -uniform. By [24]

1 |E| −2d 2 d Lemma 4.2 (or, rather, it’s proof), one has that for η = q |E| , α = qd , and N = q

3 2 2 −3d 3 2d − d |3AP (E)| = α N + ηαN = q |E| q + O q 2 .

Deﬁnition 3.3.4. Given a function f : Fq → R deﬁne

X X f(j)ν(j) = f(|y|2)m(x)m(x + y)m(x + 2y) d j∈Fq x,y∈Fq so that

1 3 2 ν(j) = (x, x + y, x + 2y) ∈ E : |y| = j . |E|3

Note that supp ν = 43(E) where

3 d 4 (E) := r ∈ Fq : there exists x, u ∈ Fq with |u| = r such thatx, x + u, x + 2u ∈ E .

k Deﬁnition 3.3.5. Deﬁne the operator Tq by k X x + y −2πi|x−y|2 k T h(x) = h(y)h e q . q 2 d y∈Fq

78 We observe that by deﬁnition, we have

1 1 X x + y −2πi|x−y|2 k ν(k) = E(x)E(y)E e q , b q |E|3 2 d x,y∈Fq

which means that 1 1 X ν(k) = E(x)T kE(x). (3.16) b q |E|3 q d x∈Fq Deﬁnition 3.3.6. We deﬁne the Gauss sum G to be

X 2 G(m, ξ) := e(2πi(x·m−ξ|x| )/q). d x,y∈Fq

One has for ξ 6= 0 that

|m|2 2πi d G(m, ξ) = e 4ξq g (ξ)

where

gd(ξ) = (u)dqd/2

for some constant u depending on p. When q ≡ 1 mod 4, u = i; for convenience we follow

[32] in working through the concrete case u = i in the calculations below.

Lemma 3.3.7. One has

X ξ Tdkf(m) = fˆ(m − ξ)fˆ(ξ)G m − , k . q 2 d ξ∈Fq

Proof. We have

X x + y 2 Tdkf(m) = q−d f(y)f e−2πi(x·m−k|x−y| )/q, q 2 d x,y∈Fq which, after a change of variables of x − y 7→ x, becomes

−d X −2πi y·m x −2πi x·m−k|x|2 /q q e q f(y)f y + e ( ) . 2 d x,y∈Fq

79 x Setting g(y) = gx(y) = f(y)f(y + 2 ), and using τt(·) to denote translation by t, we have

X x/2 ˆ ˆ ˆ 2πiξ· q gˆ(m) = f ∗ τ[x/2f(m) = f(m − ξ)f(ξ)e d ξ∈Fq so

X X ξ 2 k ˆ ˆ −2πi(x·(m− 2 )−k|x| )/q Tdq f(m) = f(m − ξ)f(ξ) e d d ξ∈Fq x∈Fq X ξ = fˆ(m − ξ)fˆ(ξ)G m − , k . 2 d ξ∈Fq

Deﬁnition 3.3.8. Deﬁne C to be the complex conjugation operator

C : f 7→ Cf := f.

Deﬁnition 3.3.9. Set

2 ! 3d+1 −6 X Y j M(q) := q |E| C Eb(mj)Eb(mj − ξj)Eb(ξj) . d j=1 m1,ξ1,m2,ξ2∈Fq , 2 2 |m1−ξ1/2| =|m2−ξ2/2| Lemma 3.3.10.

qd q2 kνk2 = M(q) + O (3.17) L2 |E|4

d/4+ and if |E| & q then 3 q −3 2 4 (E) min q, |E| |3AP (E)| . & M(q)

Proof. We have by (3.16) and Lemma 3.3.7 that

1 X ν(k) = q−1 E(x)T kE(x) b |E|3 q d x∈Fq 1 X X ξ =q−1qd Eb(m)TdkE(m) = q−1qd|E|−3 Eb(m)Eb(m − ξ)Eb(ξ)G(m − , k). |E|3 q 2 d d m∈Fq ξ,m∈Fq

80 Then

2 d −1 d d/2 1 X 2πi|m− ξ | /4qk ν(k) = (±i) q q q Eb(m)Eb(m − ξ)Eb(ξ)e 2 b |E|3 d ξ,m∈Fq so that

2 2 2 ξ1 ξ2 2 3d−2 −6 X Y j 2πi |m1− 2 | −|m2− 2 | /4qk |νb(k)| = q |E| C Eb(mj)Eb(mj − ξj)Eb(ξj) e . m1,ξ1, j=1 d m2,ξ2∈Fq This gives

X 2 |νb(k)| = d k∈Fq

3d−2 2 ! 2 2 q ξ1 ξ2 X Y j X 2πi |m1− 2 | −|m2− 2 | k/q −2 = C Eb(mi)Eb(mj − ξj)Eb(ξj) e + q |E|6 d m1,ξ1, i=1 k6=0∈Fq d m2,ξ2∈Fq 2 ! 3d−1 −6 X Y j =q |E| C Eb(mj)Eb(mj − ξj)Eb(ξj) + II d j=1 m1,m2,ξ1,ξ2∈Fq , 2 2 |m1−ξ1/2| =|m2−ξ2/2| =q−2M(q) + II where we have written

3d−2 2 ! q X Y j II := C Eb(mj)Eb(mj − ξj)Eb(ξj) |E|6 d j=1 m1,ξ1,m2,ξ2∈Fq , 2 2 |m1−ξ1/2| =|m2−ξ2/2| Applying properties of the Fourier transform and Plancherel, this will be

2 3d−2 q X = Eb(m)Eb(m − ξ)Eb(ξ) |E|6 d m,ξ∈Fq 2 3d−2 q X = Eb ∗ Eb(ξ)Eb(ξ) |E|6 d ξ∈Fq 2 3d−2 q −d X = q E2(x)E(x) |E|6 d x∈Fq q3d−2 qd−2 = q−2d|E|2 = , |E|6 |E|4

81 d which gives (3.17) since we’ve shown q2 kνk2 = M(q) + O q . b L2 |E|4 Finally, by (3.17), the support properties of ν, the Cauchy-Schwarz inequality, and

Plancherel, we have 2 2 2 1 X 1 X q−2|E|−3 |3AP (E)| = ν(0)2 = ν(j) = 43(E)ν b q q

1 3 1 X 2 ≤ 4 (E) ν q q d−2 1 3 2 1 3 −2 q = 4 (E) kνk 2 = 4 (E) q M(q) + O q b ` q |E|4 which yields the claimed lower bound on |43(E)|.

3.3.3 Circular bounds and the proof of Theorem 3.0.14

Deﬁnition 3.3.11. Set

2 Sr = x : kxk = r .

d Lemma 3.3.12 (Iosevich-Rudnev Lemma 5.1). For non-zero m ∈ Fq

−d d−1 2 Sbr(m) . q q

and

d−1 |Sr| ≈ q .

Deﬁnition 3.3.13. Deﬁne

2 X 0 0 0 0 σE(m, ξ) = Eb(m )Eb(m − ξ )Eb(ξ ). 0 0 d 0 0 2 2 m ,ξ ∈Fq ,|m −ξ /2| =|m−ξ/2| d Lemma 3.3.14. For non-zero m ∈ Fq

2 −d−1 σE(m) . q |E|.

Proof. Applying Plancherel, given any f, g

X ξ0 X f − m0 g(m0)g(m0 − ξ0)g(ξ0) = q−d fˇ(x/2)ˇg(y + 2x)ˇg(y)ˇg(x + y) 2 0 0 d d m ,ξ ∈Fq x,y∈Fq X X =q−d fˇ(0)ˇg(y)ˇg(y)ˇg(y) + q−d fˇ(x/2)ˇg(y + 2x)ˇg(y)ˇg(x + y). d d y∈Fq x6=0,y∈Fq

82 ξ0 0 0 ξ So taking g = Eb, f 2 − m = S ξ 2 m − 2 , and applying H¨older’sinequality, we have |m− 2 | for any 1 + 1 = 1 that p1 p2

2 −d X 3 −d X x σ (m, ξ) = q S 2 (0)E (y) + q S 2 − E(y + 2x)E(y)E(x + y) E b m− ξ b m− ξ | 2 | | 2 | 2 d d y∈Fq x6=0,y∈Fq

S 2 |m− ξ | −d 2 −d 3 −d−1 −d −d d−1 −d−1 ≤q |E| + q S\2 kEk p q |E| + q q q 2 |E| q |E|. d |m− ξ | L 2 . . q 2 Lp1 (ξ6=0)

taking p1 = ∞ and p2 = 1, by Lemma 3.3.12.

Proof of Theorem 3.0.14

Proof. Using the fact that E is Salem and Lemma 3.3.12, we have

M(q) 2 3d+1 −6 X Y ≤q |E| Eb(mj)Eb(mj − ξj)Eb(ξj) d j=1 m1,ξ2,m2,ξ2∈Fq , 2 2 |m1−ξ1/2| =|m2−ξ2/2| 3d+1 −6 d 2 2 −6d 3 .q |E| (m1, ξ1, m2ξ2) ∈ Fq : |m1 − ξ1| = |m2 − ξ2| q |E|

=qd|E|−3.

Applying Lemma 3.3.3, we have that

|E|3 |3AP (E)| ≈ q2d, q3d

d −3 so by this, the bound M(q) . q |E| , and Lemma 3.3.10, we have

6 3 3 −d −3 6 −2d |E| 4 (E) min q, q · |E| q · |E| · |E| q = q min 1, & q−3d

d/2 and this is & q since |E| & q .

3.3.4 Decay and the proof of Theorem 3.0.12

Lemma 3.3.15. Suppose that

2 −β σE(m) . q |E|. (3.18)

83 Then

3 −(3d+1−β) 4 (E) & q min 1, |E|q |3AP (E)| .

2 Proof. We have by the above hypothesis on σE and by Cauchy-Schwarz that

3d+1 q X M(q) = σ2 (m, ξ)Eb(m)Eb(m − ξ)Eb(ξ) |E|6 E d m,ξ∈Fq q3d+1 X q−β|E| Eb(m)Eb(m − ξ)Eb(ξ) . |E|6 d m,ξ∈Fq q3d+1 X ≤ q−β|E| Eb(m)Eb(m − ξ) |E|6 d m,ξ∈Fq q3d+1 X 2 ≤ q−β|E|qd Eb(m) , |E|6 d m∈Fq which by Plancherel is

3d+1 q X 2 = q−β|E|qd |E(m)| |E|6 d m∈Fq q3d+1 = q−β|E||E| |E|6 q3d+1−β = . |E|4

By Lemma 3.3.10 we then have

3 1 −3 2 −(3d+1−β) 4 (E) q min 1, |E| |3AP (E)| q min 1, |E|q |3AP (E)| . & M(q) &

Corollary 3.3.16. Suppose that

2d |E| |3AP (E)| & q .

Then

3 4 (E) & q. 84 Proof. This follows immediately from Lemma 3.3.15 and Lemma 3.3.14.

Proof of Theorem 3.0.12

Proof. By Lemma 3.0.11, the hypotheses of this lemma guarantee that

−c (1−α)d 2d |3AP (E)| & q q q from which it follows by the lower bound on α that

(α−c (1−α))d 2d 2d |3AP (E)| |E| & q q q q .

By Corollary 3.3.16, the result follows.

85 Chapter 4: 3APs in dense subsets of Euclidean Space

4.1 Introduction

Deﬁnition 4.1.1. A gauge function h is a left continuous, monotonic increasing function such that h(0) = 0.

The generalized Hausdorﬀ measure Hh deﬁned with respect to the gauge function h is

s 0 constructed analogously to the s-dimensional Hausdorﬀ measure H , with the premeasure ms

0 deﬁned on the semi-ring generated by Euclidean balls replaced by the premeasure mh deﬁned

0 by mh(B(x, r)) = h(r).

Given integers N1,...,Nn,..., ∈ N, define the filtration F = (Fn)n∈N by i 1 Fn = + [0, ]: i ∈ [N1 ··· Nn] . N1 ··· Nn N1 ··· Nn Definition 4.1.2. Given a positive function τ, we say that a positive measure µ is τ-split

relative to the ﬁltration F above if for inﬁnitely many n, there exist generalized arithmetic

1 progressions Pn,1 and Pn,2 in [N1 ...Nn], lying in disinct cosets of the same generalized N1···Nn

1 arithmetic progression in [N1 ··· Nn], and such that Pn,i, i = 1, 2 are contained in N1···Nn

distinct cells of Fn and µ(Pn,i) ≥ τ(n).

We are now ready to state our main theorem.

Theorem 4.1.3. Suppose that µ is a probability measure satisfying

µ(B(x, r)) ≤ h(r)µ − a.e.

86 1 where h(r) log log log log log(r−1) is a gauge function. Then either

1. supp µ contains non-trivial 3APs or

2. For all suﬃciently large N0 and all choices of Ni > N0 for all n ∈ N, set Mn =

h(1/N1 ··· Nn). For any > 0 there exists a function τ with

−2 −2 2 2 ( Mn) −2 −4( Mn) τ(n) > ( /Mn) ( Mn)

for which µ is (τ)-split relative to F.

The proof is based upon a regularity lemma which allows, at the loss of a small amount of the mass of the measure µ, to compare µ to a measure which is “almost” the result of a

Cantor construction in which at each stage the subintervals chosen all lie in the same global arithmetic progression (see the definition of “almost-flat” in Section 4.3). The 3AP counts for this new measure may be precisely counted owing to the highly rigid nature of 3AP counts between triples of long (generalized) arithmetic progressions (P0,P1,P2) (the number of 3APs with each point lying in a distinct Pi is essentially always the maximal possible size mini6=j |Pi||Pj| or else 0, owing to Bezout’s Identity (see Lemma 4.4.4)). The situation is complicated by the existence of measures, of arbitrarily large gauge dimensions, which do not contain any 3APs in their supports (see Section 4.2), but the quantitative estimates involved are sufficiently strong to yield a quantitative statement that measures with a large gauge dimension either contain (non-trivial) 3APs or else at infinitely many scales, much of their mass is concentrated within sufficiently separated distinct cosets of some generalized arithmetic progression as formalized by the property of being τ-split.

87 4.2 Large gauges without 3APs

In this section we construct a probability measure on [0, 1] satisfying an arbitrarily strong

gauge condition which nevertheless contains no non-trivial 3APs in its support. This is in

stark contrast with the discrete setting, where Roth’s original proof of Roth’s theorem showed

N that once a set A ⊂ [N] with normalized indicator function νA := A 1A satisﬁes

kνAkL∞ ≤ c log log N

it contains non-trivial 3APs.

Let (Nn)n∈N be a sequence of integers divisible by 4 and growing arbitrarily quickly; for instance, one might take

n ·2 ·· 22 n-many times. Nn = 2

Set Mn = N1 ··· Nn.

Let i 1 Fn = + [0, : i ∈ [Mn] . Mn Mn

Deﬁne the measure µn as follows: N1 µ1 = 2 1 −1 N1 + 1 −1 N1 1 , 2N1 [ 2 ] 2N1 [ 2 ]+ 2

Mn µn = 2 1Pn

−1 where Pn is the set obtained from supp µn−1 + Mn [Nn] by splitting each “subinterval” of supp µn−1 into half, and keeping the contribution from the even integers in the ﬁrst half of

[Nn], and the odd integers in [Nn] in the right half.

∗ Let µ = w − lim µn.

It is easy to see that the limiting measure µ can have no 3APs14. For at the ﬁrst scale

−1 Mn where the points of a putative 3AP in supp µ would lie in distinct subintervals of a µn, 14aside from endpoints of subintervals, which can be easily excised without aﬀecting the gauge.

88 Nn either the ﬁrst two points must lie in the set coming from 2[ 4 ] and the third point of the 3AP in the set coming from 2[ Nn ] + 1 + Nn or else the reverse situation must hold. In 4 Nn 2 either case, parity considerations forbid that the triple of points form a 3AP.

It is also easy to see that the resulting gauge function may be made arbitrarily large depending on the choice of sequence (Nn).

Our main result, Theorem 4.1.3, shows that in a sense all fractals with large gauges which do not contain 3APs must look like this example at inﬁnitely many scales.

4.3 Almost ﬂat fractals

We say that a measure µ supported on [0, 1] is ﬂat if there exist some sequence of integers

Ni ∈ N and a ﬁltration (Fn)n∈N with

1 Fn = [N1 ··· Nn] + [0, ] , N1 ··· Nn

1 arithmetic progressions Pn ⊂ [Nn] and a sequence of Fn-measurable functions µn for N1···Nn which X µn = cn1x1+···+xn

x1∈P1,...,xn∈Pn such that

w∗ µn → µ.

Deﬁnition 4.3.1. Given an increasing positive function ν and a positive function τ, an

(, ν, τ)-almost-ﬂat measure is deﬁned to be a probability measure µ supported on [0, 1] for which the following conditions hold.

1. There exist some sequence of integers Ni ∈ N and a ﬁltration (Fn)n∈N with

1 1 Fn = [N1 ··· Nn] + [0, ] ; N1 ··· Nn N1 ··· Nn

89 2. For each n ∈ N there exist Bohr sets 1 Bn,1,...Bn,l(n) ⊂ [N1 ··· Nn], N1 ··· Nn

(l(n) ≤ 1/τ(n)) of rank dn,i ≤ ν(n) and positive numbers pi such that the measure

l(n) X µn := pi1Bn,i i=1 satisﬁes

n kE (µn+1|Fn) − µnktotal variation ≤ /2 ;

3. µn(Bn,i) ≥ τ(n) for each i = 1, . . . , ν(n);

∗ 4. The sequence of measures µn converges weak to µ.

4.4 Counting 3APs in almost-ﬂat fractals

4.4.1 Counting 3APs between three progressions

Let A1,A2,A3 be three sets. Let

3AP (A1,A2,A3) = {(x0, x1, x2) ∈ A1 × A2 × A3 : xi + xj = 2xk, i 6= j 6= k 6= i}

denote the number of 3APs between the Ai.

We use the following standard lemma from elementary number theory.

Lemma 4.4.1 (Bezout’s Identity). Suppose that b0, b1 ∈ N and let

d = gcd(b0, b1).

Then there exist integers 0 ≤ x0 ≤ b0/2, 0 ≤ x1 ≤ b1/2 such that

x0b0 − x1b1 = d (4.1) and all other solutions (xe0, xe1) to (4.1) take the form b b (x , x ) = (x + n 1 , x + n 0 ). e0 e1 0 d 1 d 90 Lemma 4.4.2 (Quantitative Bezout’s Identity). Suppose that b0, b1 ∈ N and let

d = gcd(b0, b1)

and let N0,N1 ∈ N be such that bi Ni. Let m ∈ N be an integer such that m <

mini=0,1 (biNi/16).

1 Then there is a constant c = c(b0, b1,N0,N1) ∈ [ , 2] such that there are b1

N N c min 0 , 1 b1/d b0/d

solutions (x ˜0, x˜1) in [N0] × [N1] to

x˜0b0 − x˜1b1 = md.

Proof. Suppose ﬁrst that m < b1. According to Lemma 4.4.1, there exist integers

0 ≤ x0 ≤ b0/2, 0 ≤ x1 ≤ b1/2, such that

x0b0 − x1b1 = d and all other solutions (xe0, xe1) to (4.1) take the form

b b x + n 1 , x + n 0 ; 0 d 1 d

from this we obtain solutions

b b (x ˜ , x˜ ) = mx + mn 1 , mx + mn 0 (4.2) 0 1 0 d 1 d

x˜0b0 − x˜1b1 = md

91 and using the bounds on b0, b1, x0, and x1, we have that if there are S distinct such pairs in

[N0] × [N1] then

N N S = 1 + c min 0 , 1 (4.3) mb1/d mb0/d

for some c ∈ [1/4, 2].

b1 Now if m = a + k d for some a ∈ [b1] and k ∈ N then any solution to (4.2) corresponds to a solution to

y0b0 − y1b1 = ad

where y0 =x ˜0 and y1 =x ˜1 + k. Counting all such (x ˜0, x˜1) ∈ [N0] × [N1] thus corresponds to

counting those

(y0, y1) ∈ [N0] × ([N1] + k)

which we may do by (4.3): there are

N N + k N k c min 0 , 1 − c0 min 0 , ab1/d ab0/d ab1/d ab0/d

many solutions for some c, c0 ∈ [1/4, 2], which can be seen to be

N N c00 min 0 , 1 ab1/d ab0/d

in the event that kb1 ≤ mini=0,1 Nibi/8.

P Lemma 4.4.3. Let P0,P1, and P2, Pi = bij[Nij] be proper homogeneous GAPs of ranks

ri,i = 0, 1, 2 whose generators bij, i ∈ [3], j ∈ [ri] have greatest common divisor d. Suppose that max (bij) min (Nij). Then the number of solutions to

x0 + x1 − x2 = md is

≈ min |Pi||Pj| i6=j

92 provided that

m . min (Nij) .

Proof. We will treat quantities depending on the bij as constants for simplicity.

We perform an induction on r := r0 + r1 + r2. Notice that a corollary of this lemma is that if Q0,Q1 are any two proper homogeneous GAPs then the number of solutions to

y0 − y1 = m gcd (Q0,Q1) is

min (|Q0| , |Q1|) . (4.4)

We will use this corollary, for GAPs Q0,Q1 whose ranks sum to s < r + 1, to perform the

induction.

So suppose that the conclusion to the lemma holds for all GAPs whose ranks sum to less

than r + 1. Let m ∈ N be suﬃciently small. For any (x0, x1, x2) ∈ P0 × P1 × P2 the equation

x0 + x1 − x2 = md is equivalent to the expression

y0 − y1 = md (4.5)

where

y0 =x0,

and

y1 =x2 − x1. (4.6)

By the generalized Bezout identity, y = nd where d = gcd (N ,N ) , which is a 1 1 1 2j2 1j1 j1,j2

GAP Q1 of rank 1 < r1 +r2. Similarly, y0 lies in a GAP Q0 = P0 of rank r1+ < r1 +r2 +r3 =

r + 1. So we may apply induction.

93 By the inductive hypothesis, we may solve for y1 = nd1 so long as

y1 min (min) (Nij,N2j) , . j in which case there are

≈ min (|P1| , |P2|) solutions (x1, x2) to y1 = x2 − x1 in P1 × P2 by the above stated corollary.

Similarly, by the inductive hypothesis and consequent corollary we have that there are

≈ min (|Q0| , |Q1|) ≈ min (|P0| , |P1 − P2|)

solutions to (4.5) in Q0 × Q1.

Now each element of P1 − P2 can be written ≈ S ways as a combination of elements of

P1 and P2, then the corollary tells us that S ≈ min (|P1| , |P2|). Thus |P | |P | |P − P | = 1 2 1 2 S |P | |P | ≈ 1 2 min (|P1| , |P2|)

= max (|P1| , |P2|) .

Thus we have that there are

min (|P0| , |P1 − P2|) ≈ min (|P0| , max (|P1| , |P2|)) many solutions to (4.5), each of which may be written in

≈ min (|P1| , |P2|) many distinct ways, leading to a total of

≈ min (|P0| , max (|P1| , |P2|)) · min (|P1| , |P2|)

= min |Pi| |Pj| i6=j many solutions to x0 + x1 − x2 = md in P0 × P1 × P2.

94 Lemma 4.4.4. Let P0,P1, and P2 be GAPs of ranks ri, with

r −1 Xi Pi = ai + bij[Nij] j=0 where

max bij min Nij.

Then either

3AP (P0,P1,P2) ≈ min |Pi| |Pj| i6=j

or else

gcd (bij) 6 | (ai + aj − 2ak)

for all permutations (ai, aj, ak) of (a0, a1, a2).

Proof. This is immediate from Lemma 4.4.3.

4.4.2 The counting lemma

Deﬁne 2 ZZ Y Λ3(f0, f1, f2; g) := g(x, r) fi(x − ir) dx dr. i=0 Lemma 4.4.5 (Counting Lemma). Let µ be an (, ν, τ)-almost-ﬂat probability measure rel-

ative to some ﬁltration

i 1 Fn = + [0, ]: i ∈ [Nn] N1 ··· Nn N1 ··· Nn

1 and that µ(B(x, r)) ≤ r/ log log log log log r . Then either µ is τ-split for

τ(n) ≈ 1/ log Mn

there exists a constant c > 0 and a sequence of functions

2 gn : [0, 1] →[1/ log n, 1]

95 such that for δ > 0

Λ3(µn; gn; δ) = Λ3 (E (µ|Fn); gn(x, r)(1 − φδ(r)) = c − O(δ)

where φδ = 1B(0,δ) ∗ 1B(0,δ).

Proof. Let Mn = N1 ··· Nn.

Let 1 ρ = 1 − . n n2

−1 Let (Bn,i≤ln )n∈N be a sequence of Bohr sets in Mn [Mn] such that

ln X Mn (µ|F ) = w 1 + M −1[N ] E n n,i |B | Bn,i n n i=1 n,i as guaranteed by the hypothesis that µ is almost-ﬂat.

By Lemma 4.5.7 (together with Lemma 4.5.6), each Bohr set Bn,i with rank dn,i contains a generalized arithmetic progression (GAP) Pn,i with

|Pn,i| ≥ cdn,i |Bn,i|

2 −4dn,i for cdn,i = (dn,i) (Note that the entire space serves as a GAP satisfying the reverse containment with similar bounds as a direct consequence of Lemma 4.5.6.) Thus up to multiplicative terms arbitrarily close to, e.g., 100 as a function of the ranks dn,i, the Bohr sets Bn,i will have 3AP counts comparable to those of GAPs.

Without loss of generality, by pigeonholing we may assume that the GAPs

rn,i,j X Pn,i = an,i + bn,i,j[Ln,i,j] j=1 satisfying

bn,i,j Ln,i,j at the expense of slightly increasing the number, ln, of GAPs, a consequence of the gauge assumption on µ and that µ is supported in [0, 1].

96 By Lemma 4.4.4, we then have for each triple (B0,B1,B2) of Bn,i that either

3AP (B0,B1,B2) = Cd min |Bi| |Bj| i6=j

for some function Cd of the rank d = d0 + d1 + d2 of the Bi or else the GAPs Pi ⊂ Bi must

lie in at least two distinct cosets of the same progression.

This gives that, in the case that the Pi,n do not lie in distinct cosets of some progression,

Mn Mn 2 Λ3(wij ,n 1Bn,i ) = Cdwi0 wi1 wi2 min Mn. j j Bn,ij Bn,ij

Suppose now that (ηn) is a sequence of positive numbers ≤ 1 satisfying

ηn ≥ 1 − (ρ1 ··· ρn−1(1 − ρn),

and that

X Mn 2 Cdwi0 wi1 wi2 min Mn (4.7) j Bn,ij (Bn,i0 ,Bn,i1 ,Bn,i2 )Bn

X Mn 2 ≤ηn Cdwi0 wi1 wi2 min Mn (4.8) j Bn,ij (Bn,i0 ,Bn,i1 ,Bn,i2 ) where

Bn = {(Bn,i0 ,Bn,i1 ,Bn,i2 ): |3AP (Bn,i0 ,Bn,i1 ,Bn,i2 )| = 0}

holds for n = 1,...,N.

Then we have

Λ3(µN ; gN ) ≥ ρ1 ··· ρN

15 where 1/ log N ≤ gN (x, r) ≤ 1 is chosen so that

ZZ 2 X Mn 2 Y gN (x, r) Cdwi0 wi1 wi2 min Mn 1Bn,ij (x − jr) dx dr = 1. j B n,ij j=0 (Bn,i0 ,Bn,i1 ,Bn,i2 )

15 By the gauge assumption on µ, 1/ log N ≤ gN ≤ 1 and the proceeding calculations, such a gN exists.

97 It is straightforward to check that as a consequence of the rigidity of 3AP counts between

GAPs guaranteed by Lemma 4.4.4,

Λ3(µN ; gN ; δ) ≥ c − O(δ).

Finally, (4.7) follows from an assumption that µ is not τ-split, completing the lemma.

2 Deﬁnition 4.4.6. We call a sequence of functions gn : [1/ log n, 1] →[0, log n] as in Lemma

4.4.5 a renormalizing sequence for µ.

4.5 An (L1, U 2) regularity lemma

4.5.1 The regularity lemma

Lemma 4.5.1. Let G be a ﬁnite abelian group and let f : G → [0,M]. Set δ = kfkL1(G).

Let > 1/4M|G|. Suppose there exists ξ ∈ Gˆ such that |fb(ξ)| ≥ , where > 0. Then there exists θ ∈ G such that the shifted Bohr set

Bθ = B({ξ} , c/M) + θ satisﬁes

1 kfkL (Bθ) ≥ δ + c.

Here c is an absolute constant.

Proof. For ρ > 0 and θ ∈ G, deﬁne

Bθ,ρ = B({ξ} , ρ) + ξ(θ) = {x ∈ G : |ξ(x) − ξ(θ)| < ρ} .

Let δ > 1/|G|. Then there exists a subset Θ of G such that ξ(Θ) is δ-separated. Choose a maximal such Θ.

98 1 Let ρ ∈ (0, 2 ) be a constant to be speciﬁed later and

2πiθ Bθ = x ∈ G : ξ(x) − e < ρ .

Let η = sup kfk 1 − δ. For each θ, let c ∈ [−1, 1] be such that kfk 1 = δ + c η. θ L (Bθ) θ L (Bθ) θ

Then letting θi be a maximal ρ/2-separated set and denoting the Haar measure on G by m Z Z X < fˆ(ξ) = fξ dm ≈ fξ d

i Bθi ! X 1 Z = m(B ) kfk 1 ξ(θ ) + f (ξ − ξ(θ )) dm θi L (Bθ ) i i i m(Bθ ) i i Bθ9

X = m(B )(δ + c η) ξ(θ ) + O(Mρ) θi θi i i

Z X = δξ dm + O(ρ) + m(B )(c η) ξ(θ ) + O(Mρ) θi θi i i .η + Mρ.

Choosing ρ ≈ /(2M) then shows that η , so that for some θi, kfk 1 ≥ δ + c . & L (Bθi )

+ Given a Bohr set B in a ﬁnite abelian group G, for a positive function f : G → R with relative density δB = kfkL1(B), deﬁne

kf − δBkU 2(B) := f B − δB U 2(G)

where ( f on B f B = . δB on G \ B Note that in fact

kf − δBkU 2(B) = kf|B − δB1BkU 2(G) .

We have the following density increment lemma.

Lemma 4.5.2. Let G be a ﬁnite abelian group and B ⊆ G. Let f : G → [0,M] such that

f = δB := kfkL1(B)

99 on G \ B. Then

2 kf − δBkU 2(B) ≤ M max |fb(ξ)|. 06=ξ∈Gˆ

Proof. Deﬁne g = f − δB. Then g = 0 on G \ B, and so

4 4 4 2 2 2 2 kgk 2 = kgk 2 = kgk ≤ kgk kgk = kgk 2 kgk . U (B) U (G) b `4(Gˆ) b `2(Gˆ) b `∞(Gˆ) L (G) b `∞(Gˆ)

2 2 Since g ≤ M and g ≥ −δB ≥ −M, we have kgkL2(G) ≤ M . To complete the proof, it suﬃces to show that

f(ξ) if ξ is non-trivial g(ξ) = 0 if ξ is trivial

For all ξ ∈ Gˆ,

X X gb(ξ) = f(x)ξ(x) − δBξ(x). x∈G x∈G

If ξ is non-trivial, then the last sum is zero, and so gb(ξ) = fb(ξ). If ξ is the trivial character, then

X X X gb(ξ) = f(x) + δB − δB = |B|δB + |G \ B|δB − |G|δB = 0. x∈B x∈G\B x∈G

Lemma 4.5.3. Let G be a ﬁnite abelian group. Let f : G → [0,M], δG = kfkL1(G) and > 0 such that

kf − δGkU 2(G) ≥ .

Then there is a Bohr set B of rank ν ≤ −2 M and radius ρ = 2 /M such that

2 δB := kfkL1(B) ≥ δ + ν /M and

kf − δBkU 2(B) < . (4.9)

100 ˆ Proof. Applying Lemma 4.5.2 to f implies there exists a non-trivial character ξ1 ∈ G such

that

2 −1 M ≤ |fb(ξ1)|.

2 2 By Lemma 4.5.1, there is a shifted Bohr set B1 = B({ξ1} , c /M ) + θ1 such that

2 1 δB1 := kfkL (B1) ≥ δ + c /M. (4.10)

2 2 If kf − δB1 kU (B1) < , then we take B = B1, and we are done. If kf − δB1 kU (B1) ≥ , then

we continue as follows. Deﬁne

f on B1 f1 = δB1 on G \ B1

1 2 Then δB1 = kf1kL (B1) and kf1 − δB1 kU (B1) > . Applying Lemma 4.5.2 to f1 implies there ˆ exists a non-trivial character ξ2 ∈ G such that

2 −1 M ≤ |fb1(ξ2)|.

2 2 By Lemma 4.5.1, there is a shifted Bohr set B2 = B({ξ2} , c /M ) + θ2 such that

2 1 kf1kL (B2) ≥ δB1 + c /M. (4.11)

c 16 c 1 Write B2 = (B1 ∩ B2) ∪ (B1 ∩ B2). Since f1 = δB1 < kf1kL (B2) on B1 ∩ B2, it follows that

2 2 1 1 kf1kL (B1∩B2) > kf1kL (B2) ≥ δB1 + c /M ≥ δG + 2c /M.

Set f0 = f, B0 = G, δ0 = kf0kL1(G), and ν = 0 and suppose that (4.9) fails. Then

1 1 2 2 kf − δk 2 = f\− δ ≤ f\− δ f\− δ ; . 0 U (B0) 0 0 0 0 0 `4(Gb) `2(Gb) `∞(Gb)

16A minor point is that this isn’t quite right if we want to use facts about Bohr zero sets rather than look for facts about intersections of Bohr sets with diﬀerent shifts. However, the density incrememnt of f on ˆ B1 corresponds to the largeness of f at the character ξ1, and we may then ﬁnd a shift of a single Bohr set B({ξ1, ξ2} , ρ) at which f exhibits a density increment. For simplicity we will ignore this technical point.

101 hence sup fˆ(ξ) ≥ 2 / kf − δk ≥ 2 /M so there exists a non-trivial character ξ such ξ6=0 L2 1 that

2 fb(ξ1) & /M.

2 By Lemma 4.5.1, there is a shifted Bohr set B1 = B(ξ1, /M) + θ1 such that

2 kfk 1 ≥ δ + /M; L (B1)

we suppose θ chosen to maximize kfk 1 . 1 L (B1) Deﬁne ( f0 on B1 f1 = . kfk 1 on G \ B L (B1) 1 We inductively perform the following iteration, of which the above represents the base- case.

Suppose already given a sequence of functions fi, i ≤ j and shifted Bohr sets Bi =

2 2 B(ξi, /M) + θi for i < j, with kfikL1(G) =: δi ≥ δ + i /M, where ( fj−1 on Bj fj = kf k 1 on G \ B j−1 L (Bj−1) j−1

2 2 i so that fi = f on ∩i0

2 kfkL1(Bi) ≥ δ + i /M.

kfj − δjkU 2(G) . then the iteration terminates.

If not, then as in the case of f0, Lemma 4.5.1 provides a frequency ξj and shifted Bohr-set

2 Bj = B(ξj, /M) + θj such that

2 kf k 1 ≥ δ + /M; j L (Bj ) j

102 j−1 since fj ≡ δj oﬀ B , the mass responsible for this increased density must all be contained

j−1 within B (where fj ≡ f), so in fact we have found that

2 kfkL1(Bj ) ≥ δ + (j + 1) /M.

By choosing each shift θj maximal, we can guarantee that the frequencies ξ1, . . . , ξj must

be distinct.

−2 Since δν > 1 is an impossibility, this gives that (4.9) must hold after at most ν = M

iterations.

Lemma 4.5.4. [Regularity Lemma] Let G be a ﬁnite abelian group. Let f : G →[0,M],

δ = kfkL1 , A = supp(f) and let > 0. 1 δ M 2 −2 Then there are disjoint (shifted) Bohr sets B1,...,Bτ , τ ≤ of rank ν ≤ M

2 and radius ρ = /M and positive functions fB1 ,. . . ,fBν , and fsmall with

supp fBi ⊂ Bi,

kf − δ k 2 ≤ , Bi Bi U (Bi)

kfsmallkL1 ≤

R where δB = fB , such that i Bi i X f = fsmall + fBi . i≤ν

Proof. Set f0 = 0, A0 = ∅, B0 = ∅.

We apply Lemma 4.5.3 and iterate.

Suppose that disjoint sets Ai, and Bohr sets Bi ⊃ Ai as in the statement of this lemma have been deﬁned.

⊥ ⊥ Set H = (∪i

103 Set

δH := kfkL1(H)

and ( f| on H f := H ; H ⊥ δH on H

−2 Since kfH kL1(G) ≥ , applying Lemma 4.5.3 to fH yields a Bohr set Bj of rank νj ≤

2 0 and radius ρ = /M and a set Aj such that

0 2 δ := fH |A0 ≥ δH + νj ; j 1 L (Bj )

0 since fH ≡ δH oﬀ of H, setting Aj := Aj ∩ H this shows in fact that

2 2 δj := f|A 1 ≥ δH + νj ≥ +νj . j L (Bj )

We are then in a position to continue the iteration.

Suppose the iteration lasts τ steps. Then by Lemma 4.5.6 for each i ≤ τ we have the

bound

|B | 2 i ≥ 2 /Mνi ≥ 2 /M1/ ; |G|

we then calculate

2 2 X |Bi| X 2 1/ 2 1/ δ = kfk 1 ≥ kf|A k 1 ≥ /M = τ /M L (G) |G| i L (Bi) i≤τ i≤τ

so that 1 δ M 2 τ ≤ .

This completes the proof of the lemma.

4.5.2 Bohr sets

For θ ∈ = / deﬁne kθk := d(θ, 0) where d denotes the metric on . T R Z T T

104 For ρ > 0, S ⊆ Gˆ, and R : S → T, deﬁne the Bohr set B(S, R, ρ) = x ∈ G : max kξ(x) − R(ξ)k < ρ . ξ∈S T

The number ρ is called the radius of the Bohr set, the number |S| is called the rank of the

Bohr set, and the function R deﬁnes the shifts of the Bohr set.

When R ≡ 0, we call B(S, R, ρ) a Bohr0 set. Given a set of frequencies S and a radius

ρ, we will write B(S, ρ) for the Bohr0 set B(S, 0, ρ).

We will also use the following related notion.

Deﬁnition 4.5.5. A proper coset progression in an abelian group G is a subset of the form

P + H ⊂ G where P is a proper generalized arithmetic progression and |P + H| = |P ||H|.

A proper coset progression P + H is symmetric if the progression P is homogeneous, that is,

P = [−N,N] · v for some vector v.

The following facts about Bohr0 sets are standard.

Lemma 4.5.6. [Lemma 4.20 of [49]] Let B = B(S, ρ) be a Bohr0 set of rank d in G with

1 radius ρ ∈ (0, 2 ). Then |B| ≥ ρd|G|.

Lemma 4.5.7. [Lemma 4.22 of [49]] Let B = B(S, ρ) be a Bohr0 set of rank d in G with

1 radius ρ ∈ (0, 2 ). Then there exists a symmetric proper coset progression P + H of rank r ≤ d obeying the inclusions

B(S, r−2rρ) ⊂ P + H ⊂ B(S, ρ)

and with

|P | ≥ ρdd−4d2 .

105 Furthermore, H = S⊥.

4.6 Decomposing a large fractal into an almost-ﬂat portion and a small pseudorandom error

Theorem 4.6.1. Let µ be a probability measure on [0, 1] satisfying µ(B(x, r)) ≤ h(r) for

1 µ-a.e. x for a suﬃciently large gauge function h log log log log r and F be a ﬁltration

as in Section 4.5, and let > 0. Then there exists an (, ν, τ)-almost-ﬂat measure νstr also R satisfying νstr(B(x, r)) ≤ 2h(r) νstr-a.e. and a signed measure νunif satisfying dνunif = 0 and kνunif kU 2 ≤ such that

µ = νstr + νunif ,

n where setting Mn = h(1/N1 ··· Nn) and n = /2 ,

−2 −2 2 2 (n M) −2 −4(n Mn) τ(n) ≥ (n /M) (n Mn)

and 1 2 1 M n ν(n) ≤ n . n n Proof. Let

En(f) := E (f|Fn)

be the projection onto the Fn-measureable functions, and

∆n := En+1 − En.

We consider sequences of functions fn,str, fn,unif , and fn,small deﬁned as follows.

Set f1 = E1(µ). Given an Fn measurable function fn with

−n kfnkL∞ . Mn := 1/h(M )

−1 n on Mn [Mn], we set n = /2 and apply Theorem 4.5.4 to ﬁnd a decomposition

fn = fn,str + fn,small

106 where τ Xn fn,str = fn,Bn,i i=1 for 1 2 1 Mn n τn ≤ n n

−2 Bohr sets Bn,i of rank ≤ n Mn and size

−2 −2 2 2 (n M) −2 −4(n Mn) |Bn,i| ≥ (n /M) (n Mn)

where for δB := fB 2 , n,i n,i U (Bn,i)

fn,B − δB 2 ≤ n . n,i n,i U (Bn,i)

Inductively, we set Gn+1 = ∪i≤νn Bn,i and deﬁne

fn+1 = En+1(µ|Gn+1 )

on Gn+1 and Z fn+1 = fn+1/|Gn+1

c on Gn+1. We then set   X νn+1 = cn fn+1,small + δBi 1Bi  |Gn ; i≤νn+1 where cn is chosen so that νn is a probability measure. Then

n kfn − νnkU 2 ≤ /2

∗ and the νn converge weak to a measure νstr for which E (νstr|Fn) = νn.

107 4.7 Counting 3APs in large fractals

Here we prove the following theorem which says that pseudorandom subsets of almost

ﬂat fractals share their 3AP statistics.

Lemma 4.7.1. Let ν be an (, ν, τ)-almost-ﬂat probability measure relative to some ﬁltration

i 1 F = + [0, ]: i ∈ [N ] . n N n N n n

Write µn for E (µ|Fn) and νn for E (ν|Fn) and suppose that for each n ∈ N

n kµn − νnkU 2 ≤ /2 .

Then

1. If (gn) is a renormalizing sequence for ν, then

|Λ3(νn; gn; δ) − Λ3(µn; gn; δ)| ≤ .

2. If ν is τ-split and is the measure built from µ in the proof of Lemma 4.6.1 then µ is

τ-split as well.

Proof. The ﬁrst claim is a variant of a standard argument. The second follows from the proof of Lemma 4.6.1.

Proof of Theorem 4.1.3. By Lemma 4.6.1, given any ﬁltration (Fn) as in Section 4.5, there exists an (, ν, τ)-almost-ﬂat measure ν such that writing µn for E (µ|Fn) and νn for E (ν|Fn) we have that

n kµn − νnkU 2 ≤ /2 .

Thus by Lemma 4.7.1 we obtain the conclusion.

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