Related papers: On Roth's theorem on progressions

Suppose that $A \subset \{1,\dots, N\}$ has no two elements differing by a square. Then $|A| \ll N e^{-c\sqrt{\log N}}$.

Inspired by the Erd\"os-Turan conjecture we consider subsets of the natural numbers that contains infinitely many aritmetic progressions (APs) of any given length - such sets will be called AP-sets and we know due to the Green-Tao Theorem…

Let $B$ be a set of natural numbers of size $n$. We prove that the length of the longest arithmetic progression contained in the product set $B.B = \{bb'| \, b, b' \in B\}$ cannot be greater than $O(\frac{n\log^2 n}{\log \log n})$ and…

Let $k$ and $n$ be fixed positive integers. For each prime power $q\geqslant k\geqslant 3$, we show that any subset $A\subseteq \mathbb{Z}_q^n$ free of $k$-term arithmetic progressions has size $|A|\leqslant c_k(q)^n$ with a constant…

In the present paper we prove a certain lemma about the structure of "lower level-sets of convolutions", which are sets of the form $\{x \in \Z_N : 1_A*1_A(x) \leq \gamma N\}$ or of the form $\{x \in \Z_N : 1_A*1_A(x) < \gamma N\}$, where…

Let $m\geq 3$. Suppose that $$ 1-2^{-2^{m^24^m}}<\gamma<1. $$ Then the set $$ \{p\text{ prime}:\, p=[n^{\frac1\gamma}]\text{ for some }n\in{\mathbb N}\} $$ contains infinitely many non-trivial $m$-term arithmetic progressions.

We prove that if $A\subseteq \{ 1,2,\dots, N \}$ does not contain any solution to the equation $x_1+\dots+x_k=y_1+\dots+y_k$ with distinct $x_1,\dots,x_k,y_1,\dots,y_k\in A$, then $|A|\ll {k^{3/2}}N^{1/k}.$

A strictly increasing sequence of positive integers is called a slightly curved sequence with small error if the sequence can be well-approximated by a function whose second derivative goes to zero faster than or equal to $1/x^\alpha$ for…

We show that subsets of $\mathbb{F}_q^{\infty}$ of large Fourier dimension must contain three-term arithmetic progressions. This contrasts with a construction of Shmerkin of a subset of $\mathbb{R}$ of Fourier dimension $1$ with no…

In this paper, we study $k$-term arithmetic progressions $N, N+d, ..., N+(k-1)d$ of powerful numbers. Under the $abc$-conjecture, we obtain $d \gg_\epsilon N^{1/2 - \epsilon}$. On the other hand, there exist infinitely many $3$-term…

Let U(N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1}^N. By an application of the Chen-Stein method, we show that U(N)- 2 log(N)/log(2) converges in law to an extreme type (asymmetric)…

Wall's theorem on arithmetic progressions says that if $0.a_1a_2a_3\dots$ is normal, then for any $k,\ell\in \mathbb{N}$, $0.a_ka_{k+\ell}a_{k+2\ell}\dots$ is also normal. We examine a converse statement and show that if…

Given a subset of the integers of zero density, we define the weaker notion of fractional density of such a set. It is shown how this notion corresponds to that of the Hausdorff dimension of a compact subset of the reals. We then show that…

We show that if G is a finite group and A is a subset of G with no non-trivial solutions to xz=yy then |A| < |G|/(log log |G|)^c.

We consider the nonlinear system $c_1p_1^d +c_2p_2^d + \dots + c_s p_s^d = 0$ with $c_1, c_2,\dots, c_s\in\mathbb Z$ being nonzero and satisfying $c_1 +c_2 + \dots + c_s = 0$. We show that for $s\ge 2\lfloor \frac{d^2}2\rfloor+1$ and…

For k>=3 let A \subset [1,N] be a set not containing a solution to a_1 x_1+...+a_k x_k=a_1 x_{k+1}+...+a_k x_{2k} in distinct integers. We prove that there is an epsilon>0 depending on the coefficients of the equation such that every such A…

Let S be an infinite set of non-empty, finite subsets of the nonnegative integers. If p is an odd prime, let c(p) denote the cardinality of the set {T {\in} S : T {\subseteq} {1,...,p-1} and T is a set of quadratic residues (respectively,…

We show that for every positive integer $k$ there are positive constants $C$ and $c$ such that if $A$ is a subset of $\{1, 2, \dots, n\}$ of size at least $C n^{1/k}$, then, for some $d \leq k-1$, the set of subset sums of $A$ contains a…

Bourgain and Chang recently showed that any subset of $\mathbb{F}_p$ of density $\gg p^{-1/15}$ contains a nontrivial progression $x,x+y,x+y^2$. We answer a question of theirs by proving that if $P_1,P_2\in\mathbb{Z}[y]$ are linearly…

Let $(u(n))_{n\in\mathbb{N}}$ be an arithmetic progression of natural integers in base $b\in\mathbb{N}\setminus \{0,1\}$. We consider the following sequences: $s(n)=\overline{u(0)u(1)\cdots u(n) }^b$ formed by concatenating the first $n+1$…