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 translates of $d \cdot P$ that one can guarantee to find.
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.
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…
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…
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…
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); …
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…
Two-Dimensional Patterns with Distinct Differences -- Constructions, Bounds, and Maximal Anticodes
如果连接两个点的任何两条线的长度或斜率不同，则二维网格称为\emph{configuration with distinct differences}。已知这些配置具有许多应用，诸如雷达、声纳、物理对准和时间-位置同步。而不是限制点躺在一个正方形或矩形，如前所述，我们限制的配置点之间的最大距离，这样做的动机是一个新的应用程序，这样的配置在无线传感器网络中的密钥分配。我们认为配置在六边形网格以及在传统的正方形网格中，与测量的距离都在欧几里德度量，并在曼哈顿或六边形度量。我们注意到，这些配置被限制在相应的网格中的最大反码。我们在每个网格中的每个直径分类最大的反码。我们提出了上界的点的数量在这些最大的反码中所包含的具有明显差异的模式。我们的边界解决（在消极的）一个问题的Golomb和泰勒的存在的蜂窝阵列的任意大的尺寸。我们目前的建设和下界上的点的数量在配置中包含在各种二维形状（如反码）的不同差异，考虑周期性的配置，在正方形网格中的不同差异。
