Green proved an arithmetic analogue of Szemerédi's celebrated regularity lemma and used it to verify a conjecture of Bergelson, Host, and Kra which sharpens Roth's theorem on three-term arithmetic progressions in dense sets. It shows that for every subset of $\mathbb{F}_p^n$ with $n$ sufficiently large, the density of three-term arithmetic progressions with some nonzero common difference is at least the random bound (the cube of the set density) up to an additive $ε$. For a fixed odd prime $p$, we prove that the required dimension grows as an exponential tower of $p$'s of height $Θ(\log(1/ε))$. This improves both the lower and upper bound, and is the first example of a result where a tower-type bound coming from applying a regularity lemma is shown to be necessary.
Green used an arithmetic analogue of Szemerédi's celebrated regularity lemma to prove the following strengthening of Roth's theorem in vector spaces. For every $α>0$, $β<α^3$, and prime number $p$, there is a least positive integer $n_p(α,β)$ such that if $n \geq n_p(α,β)$, then for every subset of $\mathbb{F}_p^n$ of density at least $α$ there is a nonzero $d$ for which the density of three-term arithmetic progressions with common difference $d$ is at least $β$. We determine for $p \geq 19$ the tower height of $n_p(α,β)$ up to an absolute constant factor and an additive term depending only on…
Long Arithmetic Progressions in Sets with Small Sumset
设$A, B\subseteq \mathbb{Z}$是有限的非空子集，且具有$\min A=\min B=0$，令$ $δ(A,B)={\begin{array}{ll} 1 & \hbox{if} A\subseteq B, 0 & \hbox{otherwise.} If $ \max B \leq \max A \leq |A| + |B|-3 $ and \label{one}|A+B|\leq |A|+2|B|-3-δ(A,B), then we show $ A+B $ contains an arithmetic progression with difference 1 and length $ |A| + |B|-1 $. As a corollary, if \eqref{one} holds, $ \max（B）\leq \max（A）$ and either $ \gcd（A）= 1 $ or else $ \gcd（A+B）=1 $ and $| A+B| \leq 2 |A| + |B| -3 $, then $ A+B $ contains an arithmetic progression with difference 1 and length $ |A| + |B|-1 $.
Green developed an arithmetic regularity lemma to prove a strengthening of Roth's theorem on arithmetic progressions in dense sets. It states that for every $ε> 0$ there is some $N_0(ε)$ such that for every $N \ge N_0(ε)$ and $A \subset [N]$ with $|A| = αN$, there is some nonzero $d$ such that $A$ contains at least $(α^3 - ε) N$ three-term arithmetic progressions with common difference $d$. We prove that the minimum $N_0(ε)$ in Green's theorem is an exponential tower of 2s of height on the order of $\log(1/ε)$. Both the lower and upper bounds are new. It shows that the tower-type bounds that a…
We consider, over both the integers and finite fields, Szemerédi's theorem on $k$-term arithmetic progressions where the set $S$ of allowed common differences in those progressions is restricted and random. Fleshing out a line of enquiry suggested by Frantzikinakis et al, we show that over the integers, the conjectured threshold for $\mathbb{P}(d \in S)$ for Szemerédi's theorem to hold a.a.s follows from a conjecture about how so-called dual functions are approximated by nilsequences. We also show that the threshold over finite fields is different to this threshold over the integers.
We obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem, which concerns progressions with the same colour as their common difference. Such a result has been obtained independently and in much greater generality by Sanders. Using Gowers' local inverse theorem, our bound is quintuple exponential in the length of the progression. We refine this bound in the colour aspect for three-term progressions, and combine our arguments with an insight of Lefmann to obtain analogous bounds for the Ramsey numbers of certain nonlinear quadratic equations.
We consider two problems regarding arithmetic progressions in symmetric sets in the finite field (product space) model. First, we show that a symmetric set $S\subseteq\mathbb{Z}_q^n$ containing $|S|=μ\cdot q^n$ elements must contain at least $δ(q,μ)\cdot q^n\cdot 2^n$ arithmetic progressions $x,x+d,\ldots,x+(q-1)\cdot d$ such that the difference $d$ is restricted to lie in $\{0,1\}^n$. Second, we show that for prime $p$ a symmetric set $S\subseteq\mathbb{F}^n_p$ with $|S|=μ\cdot p^n$ elements contains at least $μ^{C(p)}\cdot p^{2n}$ arithmetic progressions of length $p$. This establishes that …
We determine asymptotic growth rates for lengths of monochromatic arithmetic progressions in certain automatic sequences. In particular, we look at (one-sided) fixed points of aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue--Morse and Rudin--Shapiro substitutions, respectively. For such infinite words, we show that there exists a subsequence $\left\{d_n\right\}$ of differences along which the maximum length $A(d_n)$ of a monochromatic arithmetic progression (with fixed difference $d_n$) grows at least polynomially in $d_n$. Explicit u…
For a polynomial progression $$(x,\; x+P_1(y),\; \ldots,\; x+P_{t}(y)),$$ we define four notions of complexity: Host-Kra complexity, Weyl complexity, true complexity and algebraic complexity. The first two describe the smallest characteristic factor of the progression, the third one refers to the smallest-degree Gowers norm controlling the progression, and the fourth one concerns algebraic relations between terms of the progressions. We conjecture that these four notions are equivalent, which would give a purely algebraic criterion for determining the smallest Host-Kra factor or the smallest G…
We provide upper bounds for the size of subsets of finite fields lacking the polynomial progression $$ x, x+y, ..., x+(m-1)y, x+y^m, ..., x+y^{m+k-1}.$$ These are the first known upper bounds in the polynomial Szemerédi theorem for the case when polynomials are neither linearly independent nor homogeneous of the same degree. We moreover improve known bounds for subsets of finite fields lacking arithmetic progressions with a difference coming from the set of $k$-th power residues, i.e. configurations of the form $$ x, x+y^k, ..., x+(m-1)y^k.$$ Both results follow from an estimate of the number …
Given a subset of real numbers $A$ with small product $AA$ we obtain a new upper bound for the additive energy of $A$. The proof uses a natural observation that level sets of convolutions of the characteristic function of $A$ have small product with $A$.
For a subset $A \subseteq [N]$, we define the representation function $ r_{A-A}(d) := \#\{(a,a') \in A \times A : d = a - a'\}$ and define $M_D(A) := \max_{1 \leq d < D} r_{A-A}(d)$ for $D>1$. We study the smallest possible value of $M_D(A)$ as $A$ ranges over all possible subsets of $[N]$ with a given size. We give explicit asymptotic expressions with constant coefficients determined for a large range of $D$. We shall also see how this problem connects to a well-known problem about generalized Sidon sets.
Which finite sets $P \subseteq \mathbb{Z}^r$ with $|P| \ge 3$ have the following property: for every $A \subseteq [N]^r$, there is some nonzero integer $d$ such that $A$ contains $(α^{|P|} - o(1))N^r$ translates of $d \cdot P = \{d p : p \in P\}$, where $α= |A|/N^r$? Green showed that all 3-point $P \subseteq \mathbb{Z}$ have the above property. Green and Tao showed that 4-point sets of the form $P = \{a, a+b, a+c, a+b+c\} \subseteq \mathbb{Z}$ also have the property. We show that no other sets have the above property. Furthermore, for various $P$, we provide new upper bounds on the number of …
For a subset $A$ of $\{1,2,\ldots,N\}^2$ of size $αN^2$ we show existence of $(m,n)\neq(0,0)$ such that the set $A$ contains at least $(α^3 - o(1))N^2$ triples of points of the form $(a,b)$, $(a+m,b+n)$, $(a-n,b+m)$. This answers a question by Ackelsberg, Bergelson, and Best from arXiv:2101.02811. The same approach also establishes the corresponding result for compact abelian groups. Furthermore, for a finite field $\mathbb{F}_q$ we comment on exponential smallness of subsets of $(\mathbb{F}_q^n)^2$ that avoid the aforementioned configuration. The proofs are minor modifications of the existing…
Motivated by problems on random differences in Szemerédi's theorem and on large deviations for arithmetic progressions in random sets, we prove upper bounds on the Gaussian width of point sets that are formed by the image of the $n$-dimensional Boolean hypercube under a mapping $ψ:\mathbb{R}^n\to\mathbb{R}^k$, where each coordinate is a constant-degree multilinear polynomial with 0-1 coefficients. We show the following applications of our bounds. Let $[\mathbb{Z}/N\mathbb{Z}]_p$ be the random subset of $\mathbb{Z}/N\mathbb{Z}$ containing each element independently with probability $p$. $\bulle…
The $3k-4$ Theorem is a classical result which asserts that if $A,\,B\subseteq \mathbb Z$ are finite, nonempty subsets with \begin{equation}\label{hyp}|A+B|=|A|+|B|+r\leq |A|+|B|+\min\{|A|,\,|B|\}-3-δ,\end{equation} where $δ=1$ if $A$ and $B$ are translates of each other, and otherwise $δ=0$, then there are arithmetic progressions $P_A$ and $P_B$ of common difference such that $A\subseteq P_A$, $B\subseteq P_B$, $|B|\leq |P_B|+r+1$ and $|P_A|\leq |A|+r+1$. It is one of the few cases in Freiman's Theorem for which exact bounds on the sizes of the progressions are known. The hypothesis above is …
For a compact abelian group $G$, a corner in $G \times G$ is a triple of points $(x,y)$, $(x,y+d)$, $(x+d,y)$. The classical corners theorem of Ajtai and Szemerédi implies that for every $α> 0$, there is some $δ> 0$ such that every subset $A \subset G \times G$ of density $α$ contains a $δ$ fraction of all corners in $G \times G$, as $x,y,d$ range over $G$. Recently, Mandache proved a "popular differences" version of this result in the finite field case $G = \mathbb F_p^n$, showing that for any subset $A \subset G \times G$ of density $α$, one can fix $d \neq 0$ such that $A$ contains a large …
We establish analogues for trees of results relating the density of a set $E \subset \mathbb{N}$, the density of its set of popular differences, and the structure of $E$. To obtain our results, we formalise a correspondence principle of Furstenberg and Weiss which relates combinatorial data on a tree to the dynamics of a Markov process. Our main tools are Kneser-type inverse theorems for sets of return times in measure-preserving systems. In the ergodic setting we use a recent result of the first author with Björklund and Shkredov and a stability-type extension (proved jointly with Shkredov); …
The following combinatorial conjecture arises naturally from recent ergodic-theoretic work of Ackelsberg, Bergelson, and Best. Let $M_1$, $M_2$ be $k\times k$ integer matrices, $G$ be a finite abelian group of order $N$, and $A\subseteq G^k$ with $|A|\geαN^k$. If $M_1$, $M_2$, $M_1-M_2$, and $M_1+M_2$ are automorphisms of $G^k$, is it true that there exists a popular difference $d \in G^k\setminus\{0\}$ such that \[\#\{x \in G^k: x, x+M_1d, x+M_2d, x+(M_1+M_2)d \in A\} \ge (α^4-o(1))N^k.\] We show that this conjecture is false in general, but holds for $G = \mathbb{F}_p^n$ with $p$ an odd prim…
