An analog of the Furstenberg-Katznelson-Weiss theorem on triangles in sets of positive density in finite field geometries

A short proof of the multidimensional Szemerédi theorem in the primes

Tao conjectured that every dense subset of $\mathcal{P}^d$, the $d$-tuples of primes, contains constellations of any given shape. This was very recently proved by Cook, Magyar, and Titichetrakun and independently by Tao and Ziegler. Here we give a simple proof using the Green-Tao theorem on linear equations in primes and the Furstenberg-Katznelson multidimensional Szemerédi theorem.

Ergodic-theoretic implementations of the Roth density-increment argument

We exhibit proofs of two ergodic-theoretic results in the study of multiple recurrence using an analog of the density-increment argument of Roth and Gowers: Furstenberg's Multiple Recurrence Theorem (which implies Szemerédi's Theorem), and a two-dimensional special case of Furstenberg and Katznelson's multidimensional version of this theorem. The second of these requires also an analog of some recent finitary work by Shkredov. Many proofs of these multiple recurrence theorems are now known, but our main goal is to shed some further light on the heuristic correspondence principle that has grown…

Deducing the Density Hales-Jewett Theorem from an infinitary removal lemma

We offer a new proof of Furstenberg and Katznelson's density version of the Hales-Jewett Theorem: For any $δ> 0$ there is some $N_0 \geq 1$ such that whenever $A \subseteq [k]^N$ with $N \geq N_0$ and $|A|\geq δk^N$, $A$ contains a \textbf{combinatorial line}: that is, for some $I \subseteq [N]$ nonempty and $w_0 \in [k]^{[N]\setminus I}$ we have A \supseteq \{w: w|_{[N]\setminus I} = w_0, w|_I = \rm{const.}\}. Following Furstenberg and Katznelson, we first show that this result is equivalent to a `multiple recurrence' assertion for a class of probability measures enjoying a certain kind of st…

Monoid actions and ultrafilter methods in Ramsey theory

First, we prove a theorem on dynamics of actions of monoids by endomorphisms of semigroups. Second, we introduce algebraic structures suitable for formalizing infinitary Ramsey statements and prove a theorem that such statements are implied by existence of appropriate homomorphisms between the algebraic structures. We make a connection between the two themes above, which allows us to prove some general Ramsey theorems for sequences. We give a new proof of the Furstenberg--Katznelson Ramsey theorem; in fact, we obtain a version of this theorem that is stronger than the original one. We answer i…

Nilfactors of R^m-actions and configurations in sets of positive upper density in R^m

We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. Let E be a measurable subset of R^m, with positive upper density. Let V={0,v_1,...,v_k} be a subset of R^m. We show that for r large enough, we can find an isometric copy of rV arbitrarily close to E. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss showing a similar property for m=k=2.

Sets in $\mathbb{R}^d$ with slow-decaying density that avoid an unbounded collection of distances

For any $d\in \mathbb{N}$ and any function $f:(0,\infty)\to [0,1]$ with $f(R)\to 0$ as $R\to \infty$, we construct a set $A \subseteq \mathbb{R}^d$ and a sequence $R_n \to \infty$ such that $\|x-y\| \neq R_n$ for all $x,y\in A$ and $μ(A\cap B_{R_n})\geq f(R_n)μ(B_{R_n})$ for all $n\in \mathbb{N}$, where $B_R$ is the ball of radius $R$ centered at the origin and $μ$ is Lebesgue measure. This construction exhibits a form of sharpness for a result established independently by Furstenberg-Katznelson-Weiss, Bourgain, and Falconer-Marstrand, and it generalizes to any metric induced by a norm on $\ma…

Multiple recurrence and the structure of probability-preserving systems

In 1975 Szemerédi proved the long-standing conjecture of Erdős and Turán that any subset of $\bbZ$ having positive upper Banach density contains arbitrarily long arithmetic progressions. Szemerédi's proof was entirely combinatorial, but two years later Furstenberg gave a quite different proof of Szemerédi's Theorem by first showing its equivalence to an ergodic-theoretic assertion of multiple recurrence, and then bringing new machinery in ergodic theory to bear on proving that. His ergodic-theoretic approach subsequently yielded several other results in extremal combinatorics, as well as revea…

Ramsey Theory for Words Representing Rationals

Ramsey theory for words over a finite alphabet was unified in the work of Carlson and Furstenberg-Katznelson. Carlson, in the same work, outlined a method to extend the theory for words over an infinite alphabet, but subject to a fixed dominating principle, proving in particular an Ellentuck version, and a corresponding Ramsey theorem for k=1. In the present work we develop in a systematic way a Ramsey theory for words (in fact for ω-Z*-located words) over a doubly infinite alphabet extending Carlson's approach (to countable ordinals and Schreier-type families), and we apply this theory, explo…

$\Delta^*$ 集的局部动力结构：基于新型 Furstenberg 族代数

The local dynamical structure of $Δ^*$ sets via a new Furstenberg family algebra

本文通过引入新型 Furstenberg 族代数，证明与交换半群所有差子集相交的集合具有局部 Bohr 结构，推广了整数上的经典结果，并在 Lean 中完成形式化验证。

Three Dimensional Corners: A Box Norm Proof

In an additive group (G,+), a three-dimensional corner is the four points g, g+d(1,0,0), g+d(0,1,0), g+d(0,0,1), where g is in G^3, and d is a non-zero element of G. The Ramsey number of interest is R_3(G) the maximal cardinality of a subset of G^3 that does not contain a three-dimensional corner. Furstenberg and Katznelson have shown R_3(Z_N) is little-o of N^3, and in fact the corresponding result holds in all dimensions, a result that is a far reaching extension of the Szemeredi Theorem. We give a new proof of the finite field version of this fact, a proof that is a common generalization of…

Remarks on a Ramsey theory for trees

Extending Furstenberg's ergodic theoretic proof for Szemerédi's theorem on arithmetic progressions, Furstenberg and Weiss (2003) proved the following qualitative result. For every d and k, there exists an integer N such that no matter how we color the vertices of a complete binary tree T_N of depth N with k colors, we can find a monochromatic replica of T_d in T_N such that (1) all vertices at the same level in T_d are mapped into vertices at the same level in T_N; (2) if a vertex x of T_d is mapped into a vertex y in T_N, then the two children of x are mapped into descendants of the the two c…

Ramsey monoids

Recently, Solecki introduced the notion of Ramsey monoid to produce a common generalization to theorems such as Hindman's theorem, Carlson's theorem, and Gowers' FIN$_k$ theorem. He proved that an entire class of finite monoids is Ramsey. Here we improve this result, enlarging this class and finding a simple algebraic characterization of finite Ramsey monoids. We extend in a similar way a result of Solecki regarding a second class of monoids connected to the Furstenberg-Katznelson Ramsey Theorem. The results obtained suggest a possible connection with Schützenberger's theorem and finite automa…

Arithmetic subtrees in large subsets of products of trees

Furstenberg-Weiss have extended Szemerédi's theorem on arithmetic progressions to trees by showing that a large subset of the tree contains arbitrarily long arithmetic subtrees. We study higher dimensional versions that analogously extend the multidimensional Szemerédi theorem by demonstrating the existence of certain arithmetic structures in large subsets of a cartesian product of trees.

Approximate Euclidean Ramsey theorems

According to a classical result of Szemerédi, every dense subset of $1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large enough. Its analogue in higher dimensions due to Fürstenberg and Katznelson says that every dense subset of $\{1,2,...,N\}^d$ contains an arbitrary large grid, if $N$ is large enough. Here we generalize these results for separated point sets on the line and respectively in the Euclidean space: (i) every dense separated set of points in some interval $[0,L]$ on the line contains an arbitrary long approximate arithmetic progression, if $L$ is large en…

Schreier Sets in Ramsey Theory

We show that Ramsey theory, a domain presently conceived to guarantee the existence of large homogeneous sets for partitions on k-tuples of words (for every natural number k) over a finite alphabet, can be extended to one for partitions on Schreier-type sets of words (of every countable ordinal). Indeed, we establish an extension of the partition theorem of Carlson about words and of the (more general) partition theorem of Furstenberg-Katznelson about combinatorial subspaces of the set of words (generating from k-tuples of words for any fixed natural number k) into a partition theorem about co…

幂零系统中的同时逼近与回归时间集的乘法厚性

Simultaneous approximation in nilsystems and the multiplicative thickness of return-time sets

本文研究极小幂零系统中的同时逼近稠密性，揭示回归时间集与 nil-Bohr 集的乘法厚性，并据此推广多重回归定理、增强 van der Waerden 定理。

A new and simpler noncommutative central sets theorem

Using dynamics, Furstenberg defined the concept of a central subset of positive integers and proved several powerful combinatorial properties of central sets. Later using the algebraic structure of the Stone-Čech compactification, Bergelson and Hindman, with the assistance of B. Weiss, generalized the notion of a central set to any semigroup and extended the most important combinatorial property of central sets to the central sets theorem. Currently the most powerful formulation of the central sets theorem is due to De, Hindman, and Strauss in [3, Corollary 3.10]. However their formulation of …

Density Hales-Jewett and Moser numbers

For any $n \geq 0$ and $k \geq 1$, the \emph{density Hales-Jewett number} $c_{n,k}$ is defined as the size of the largest subset of the cube $[k]^n$ := $\{1,...,k\}^n$ which contains no combinatorial line; similarly, the Moser number $c'_{n,k}$ is the largest subset of the cube $[k]^n$ which contains no geometric line. A deep theorem of Furstenberg and Katznelson shows that $c_{n,k}$ = $o(k^n)$ as $n \to \infty$ (which implies a similar claim for $c'_{n,k}$); this is already non-trivial for $k = 3$. Several new proofs of this result have also been recently established. Using both human and com…

Direct and inverse results for popular differences in trees of positive dimension

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); …

动态 syndetic 集与 syndetic、幂等滤子的组合学

Dynamically syndetic sets and the combinatorics of syndetic, idempotent filters

本文证明动态中心 syndetic 集恰好是 syndetic、幂等滤子的成员，从而给出 Furstenberg 中心集局部超滤子刻画的全局类比，并据此解决 Host、Kra 和 Maass 提出的两个关于逐点拓扑回复集的公开问题。

Ramsey theory for p-quasicyclic groups with a view towards topological dynamics

We prove additive and multiplicative partition theorems, obtaining combinatorial results for p-quasicyclic groups, where p is a prime number. We also get density results for p-quasicyclic groups via left Følner sequences of non-empty finite subsets of it, giving a sufficient condition in order a subset of a p-quasicyclic group to contain arbitrary long arithmetic progressions. Finally, we introduce the notion of a dynamical system over p-quasicyclic groups extending the classical notion of a topological dynamical system and we prove (multiple) recurrent results for the p-quasicyclic groups. In…

A note on configurations in sets of positive density which occur at all large scales

Furstenberg, Katznelson and Weiss proved in the early 1980s that every measurable subset of the plane with positive density at infinity has the property that all sufficiently large real numbers are realised as the Euclidean distance between points in that set. Their proof used ergodic theory to study translations on a space of Lipschitz functions corresponding to closed subsets of the plane, combined with a measure-theoretical argument. We consider an alternative dynamical approach in which the phase space is given by the set of measurable functions from $\mathbb{R}^d$ to $[0,1]$, which we vie…

多维Szemerédi定理的一个近似计数版本

An Approximate Counting Version of the Multidimensional Szemerédi Theorem

本文证明了多维Szemerédi定理的一个近似计数版本，利用Behrend构造的多维扩展和超图容器引理，推广了关于无特定模式集合数量的已有结果。

向量空间拉姆齐数与弱Sidorenko仿射构型

Vector space Ramsey numbers and weakly Sidorenko affine configurations

该研究通过计数有限域向量空间子集间的仿射同态，推导仿射极值数的新界与超饱和结果，并将其与向量空间拉姆齐数关联，改进了多组二元、三元域上拉姆齐数的已知最优上界。

Combinatorial Structures on van der Waerden sets

In this paper we provide two results. The first one consists an infinitary version of the Furstenberg-Weiss Theorem. More precisely we show that every subset $A$ of a homogeneous tree $T$ such that $\frac{|A\cap T(n)|}{|T(n)|}\geqδ$, where T(n) denotes the $n$-th level of $T$, for all $n$ in a van der Waerden set, for some positive real $δ$, contains a strong subtree having a level sets which forms a van der Waerden set. The second result is the following. For every sequence $(m_q)_{q}$ of positive integers and for every real $0<δ\\leq1$, there exists a sequence $(n_q)_{q}$ of positive integer…

具有一个大加数的离散和集

Discrete sumsets with one large summand

本文研究离散阿贝尔群中至少一个加数具有正上Banach密度的和集问题，通过建立离散与紧致阿贝尔群中和集及卷积水平集之间的强对应关系，改进了前人结论，且证明避免了保测动力学和非标准分析，适用于任意基数的离散阿贝尔群。

研究 Hindman、Ramsey 和 van der Waerden 空间的一种统一方法

A unified approach to Hindman, Ramsey and van der Waerden spaces

本文提出统一框架处理 IP-收敛、R-收敛与普通收敛及其对应空间，证明一般存在性定理并构造多种反例，还用 Katětov 序刻画不同类型空间的分离条件。

双侧线性哈希与光滑格覆盖的二次密度界

Two-sided linear hashing and quadratic density bounds for smooth lattice coverings

本研究提出双侧线性哈希方法，通过随机线性投影的纤维大小界，将光滑格覆盖的复杂度从 O(n^3) 改进至 O(n^2)，并证明该阶数最优。

和集中的 Bohr 集 II：可数阿贝尔群

Bohr sets in sumsets II: countable abelian groups

本文研究可数阿贝尔群中三重和集的 Bohr 集存在性，证明三类正密度与有限划分情形下的定量结果，并推广和加强多项已知定理。