Related papers: Four-term progression free sets with three-term pr…

Croot, Lev and Pach used a new polynomial technique to give a new exponential upper bound for the size of three-term progression-free subsets in the groups $(\mathbb Z _4)^n$. The main tool in proving their striking result is a simple lemma…

We extend two well-known results in additive number theory, S\'ark\"ozy's theorem on square differences in dense sets and a theorem of Green on long arithmetic progressions in sumsets, to subsets of random sets of asymptotic density 0. Our…

Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise syndetic set contains k length arithmetic progression and such collection is also piecewise syndetic in Z: They used algebraic structure of beta N. The above result…

Let $AP_k=\{a,a+d,\ldots,a+(k-1)d\}$ be an arithmetic progression. For $\epsilon>0$ we call a set $AP_k(\epsilon)=\{x_0,\ldots,x_{k-1}\}$ an $\epsilon$-approximate arithmetic progression if for some $a$ and $d$, $|x_i-(a+id)|<\epsilon d$…

We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic…

Green and Tao famously proved in 2005 that any subset of the primes of fixed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that in fact any subset of the primes of relative density tending to…

Combining Freiman's theorem with Balog-Szemeredi-Gowers theorem one can show that if an additive set has large additive energy, then a large piece of the set is contained in a generalized arithmetic progression of small rank and size. In…

We provide a short proof of a recent result of Elkin in which large subsets of the integers 1 up to N free of 3-term progressions are constructed.

Szemeredi's regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemeredi's regularity lemma in the context of abelian groups and use it to derive some…

Let $G$ be a finite Abelian group. For a subset $S \subseteq G$, let $T_3(S)$ denote the number of length three arithemtic progressions in $S$ and Prob[$S$] $= \frac{1}{|S|^2}\sum_{x,y \in S} 1_S(x+y)$. For any $q \ge 1$ and $\alpha \in…

In this note, we show the mixing of three-term progressions $(x, xg, xg^2)$ in every finite quasirandom groups, fully answering a question of Gowers. More precisely, we show that for any $D$-quasirandom group $G$ and any three sets $A_1,…

Let $X_k$ denote the number of $k$-term arithmetic progressions in a random subset of $\mathbb{Z}/N\mathbb{Z}$ or $\{1, \dots, N\}$ where every element is included independently with probability $p$. We determine the asymptotics of $\log…

We study the repetition of patches in self-affine tilings in R^d. In particular, we study the existence and non-existence of arithmetic progressions. We first show that an arithmetic condition of the expansion map for a self-affine tiling…

Ellenberg and Gijswijt gave recently a new exponential upper bound for the size of three-term arithmetic progression free sets in $({\mathbb Z_p})^n$, where $p$ is a prime. Petrov summarized their method and generalized their result to…

A famous theorem of Szemer\'edi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a…

It is shown that for a finite set $A$ of four or more complex numbers, the cardinality of the set $C[A]$ of all cross-ratios generated by quadruples of pair-wise distinct elements of $A$ is $|C[A]|\gg…

Recently a new class of continued fraction algorithms, the $(N,\alpha$)-expansions, was introduced for each $N\in\mathbb{N}$, $N\geq 2$ and $\alpha \in (0,\sqrt{N}-1]$. Each of these continued fraction algorithms has only finitely many…

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 $(\alpha^{|P|} - o(1))N^r$ translates of $d \cdot P = \{d p :…

Celebrated theorems of Roth and of Matou\v{s}ek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $\Theta(n^{1/4})$. We study the analogous problem in the $\mathbb{Z}_n$ setting.…

We study subsets of $\mathbb{F}_p^n$ that do not contain progressions of length $k$. We denote by $r_k(\mathbb{F}_p^n)$ the cardinality of such subsets containing a maximal number of elements. In this paper we focus on the case $k=p$ and…