Related papers: Avoiding a star of three-term arthmetic progressio…

We give a self-contained exposition of the recent remarkable result of Kelley and Meka: if $A\subseteq \{1,\ldots,N\}$ has no non-trivial three-term arithmetic progressions then $\lvert A\rvert \leq \exp(-c(\log N)^{1/12})N$ for some…

Let a be a real number between 0 and 1. Ernie Croot showed that the quantity \max_A #(3-term arithmetic progressions in A)/p^2, where A ranges over all subsets of Z/pZ of size at most a*p, tends to a limit as p tends to infinity through…

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 prove a version of Szemeredi's regularity lemma for subsets of a typical random set in F_p^n. As an application, a result on the distribution of three-term arithmetic progressions in sparse sets is discussed.

Let $r_k(n)$ denote the maximum cardinality of a set $A \subset \{1,2, \dots, n \}$ such that $A$ does not contain a $k$-term arithmetic progression. In this paper, we give a method of constructing such a set and prove the lower bound…

The problem of looking for subsets of the natural numbers which contain no 3-term arithmetic progressions has a rich history. Roth's theorem famously shows that any such subset cannot have positive upper density. In contrast, Rankin in 1960…

In 1946, Behrend gave a construction of dense finite sets of integers that do not contain 3-term arithmetic progressions. In 1961, Rankin generalized Behrend's construction to sets avoiding k-term arithmetic progressions, and in 2008 Elkin…

Let A be a finite set of integers. We prove that if |A| is at least 2 and |A+A| is 3|A|-3, then one of the following is true: 1. A is a bi-arithmetic progression; 2. A+A contains an arithmetic progression of length 2|A|-1; 3. |A| is 6 and A…

In a recent breakthrough Kelley and Meka proved a quasipolynomial upper bound for the density of sets of integers without non-trivial three-term arithmetic progressions. We present a simple modification to their method that strengthens…

Addressing a question of Cameron and Erd\Ho s, we show that, for infinitely many values of $n$, the number of subsets of $\{1,2,\ldots, n\}$ that do not contain a $k$-term arithmetic progression is at most $2^{O(r_k(n))}$, where $r_k(n)$ is…

A cap set in $\mathbb{F}_3^n$ is a subset that contains no three elements adding to 0. Building on a construction of Edel, a recent paper of Tyrrell gave the first improvement to the lower bound for a size of a cap set in two decades…

We develop recent ideas of Elsholtz, Proske, and Sauermann to construct denser subsets of $\{1,\dots,N\}$ that lack arithmetic progressions of length $3$. This gives the first quasipolynomial improvement since the original construction of…

Fix a density d in (0,1], and let F_p^n be a finite field, where we think of p fixed and n tending to infinity. Let S be any subset of F_p^n having the minimal number of three-term progressions, subject to the constraint |S| is at least…

Let H stand for the set of homeomorphisms on [0,1]. We prove the following dichotomy for Borel subsets A of [0,1]: either there exists a homeomorphism f in H such that the image f(A) contains no 3-term arithmetic progressions; or, for every…

A subset $S$ of an abelian group $G$ is called $3$-$\mathrm{AP}$ free if it does not contain a three term arithmetic progression. Moreover, $S$ is called complete $3$-$\mathrm{AP}$ free, if it is maximal w.r.t. set inclusion. One of the…

For a prime $p$, a restricted arithmetic progression in $\mathbb{F}_p^n$ is a triplet of vectors $x, x+a, x+2a$ in which the common difference $a$ is a non-zero element from $\{0,1,2\}^n$. What is the size of the largest $A\subseteq…

We use topological ideas to show that, assuming the conjecture of Erd\"(o)s on subsets of positive integers having no $p$ terms in arithmetic progression (A. P.), there must exist a subset $M_p$ of positive integers with no $p$ terms in A.…

We construct subsets of {1,...,N} of cardinality at least N exp(-C(log N)^{1/(k+1)}) which do not contain arithmetic progressions of length 2^k+1. This extends a result of Behrend (1946) concerning sets which do not contain aritmetic…

In this paper we prove a basic theorem which says that if f : F_p^n -> [0,1] has the property that ||f^||_(1/3) is not too ``large''(actually, it also holds for quasinorms 1/2-\delta in place of 1/3), and E(f) = p^{-n} sum_m f(m) is not too…

We show that if A is a subset of {1,...,N} contains no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{1-o(1)} N). The approach is somewhat different from that used in arXiv:1007.5444.