Related papers: New bounds for the Furstenberg-S\'ark\"ozy theorem

Suppose that $A \subset \{1,\dots, N\}$ has no two elements differing by $p-1$, $p$ prime. Then $|A| \ll N^{1 - c}$.

We establish upper bounds on the size of the largest subset of $\{1,2,\dots,N\}$ lacking nonzero differences of the form $h(p_1,\dots,p_{\ell})$, where $h\in \mathbb{Z}[x_1,\dots,x_{\ell}]$ is a fixed polynomial satisfying appropriate…

We show that if $A\subset \{1,\ldots,N\}$ has no solutions to $a-b=n^2$ with $a,b\in A$ and $n\geq 1$ then \[|A|\ll \frac{N}{(\log N)^{c\log\log \log N}}\] for some absolute constant $c>0$. This improves upon a result of…

The Furstenberg-S\'ark\"ozy theorem asserts that the difference set $E-E$ of a subset $E \subset \mathbb{N}$ with positive upper density intersects the image set of any polynomial $P \in \mathbb{Z}[n]$ for which $P(0)=0$. Furstenberg's…

We show that if $h\in \mathbb{Z}[x]$ is a polynomial of degree $k \geq 2$ such that $h(\mathbb{N})$ contains a multiple of $q$ for every $q\in \mathbb{N}$, known as an $\textit{intersective polynomial}$, then any subset of $\{1,2,\dots,N\}$…

We show that if the difference of two elements of a set $A \subseteq [N]$ is never one less than a prime number, then $|A| = O (N \exp (-c (\log N)^{1/3}))$ for some absolute constant $c>0$.

We introduce the notion of intersective polynomials having coefficients in the ring of integers $\mathscr{O}_K$ of a number field $K$, and define a notion of upper density of subsets of $\mathscr{O}_K$. We prove that given any intersective…

Denote by $\mathcal{R}_p$ the set of all quadratic residues in $\mathbf{F}_p$ for each prime $p$. A conjecture of A. S\'ark\"ozy asserts, for all sufficiently large $p$, that no subsets $\mathcal{A},\mathcal{B}\subseteq\mathbf{F}_p$ with…

Green showed that, conditional on GRH, a subset $A \subseteq [N]$ with $\mid A \mid \gg_{\epsilon} N^{\frac{11}{12}+\epsilon}$ must contain two elements whose difference is $p-1$ for $p$ a prime. We prove an analogous unconditional result…

Let $k\geq1$ be a fixed integer, and $\mathcal P_N$ be the set of primes no more than $N$. We prove that if a set $\mathcal A\subset\mathcal P_N$ contains no patterns $p_1,p_1+(p_2-1)^k$, where $p_1,p_2$ are prime numbers, then \[…

A corner is a set of three points in $\mathbf{Z}^2$ of the form $(x, y), (x + d, y), (x, y + d)$ with $d \neq 0$. We show that for infinitely many $N$ there is a set $A \subset [N]^2$ of size $2^{-(c + o(1)) \sqrt{\log_2 N}} N^2$ not…

For every $\beta\in(0,\infty)$, $\beta\neq 1$ we prove that a positive measure subset $A$ of the unit square contains a point $(x_0,y_0)$ such that $A$ nontrivially intersects curves $y-y_0 = a (x-x_0)^\beta$ for a whole interval…

We make quantitative improvements to recently obtained results on the structure of the image of a large difference set under certain quadratic forms and other homogeneous polynomials. Previous proofs used deep results of Benoist-Quint on…

Very recently, Green and Sawhney obtained a quasipolynomial bound in the Furstenberg--S\'ark\"ozy theorem for square differences by proving an ''arithmetic level-$d$'' inequality, thereby yielding a greatly improved density increment…

We prove that every subset of $\{1,\dots, N\}$ which does not contain any solutions to the equation $x+y+z=3w$ has at most $\exp(-c(\log N)^{1/5+o(1)})N$ elements, for some $c>0$. This theorem improves upon previous estimates. Additionally,…

We prove that if $A\subset \{1,\dots,N\}$ has no nontrivial three-term arithmetic progressions, then $|A|\leq \exp(-c\log(N)^{1/6}\log\log(N)^{-1})N$ for some absolute constant $c>0$. To obtain this bound, we use an iterated variant of the…

We prove that any subset $A \subseteq [3]^n$ with $3^{-n}|A| \ge (\log\log\log\log n)^{-c}$ contains a combinatorial line of length $3$, i.e., $x, y, z \in A$, not all equal, with $x_i=y_i=z_i$ or $(x_i,y_i,z_i)=(0,1,2)$ for all $i = 1, 2,…

We prove that for some universal $c$, a non-collinear set of $N>\frac{1}{c}$ points in the Euclidean plane determines at least $c \frac{N}{\log N}$ distinct areas of triangles with one vertex at the origin, as well as at least $c…

We prove new combinatorial results about polynomial configurations in large subsets of finite fields. Bergelson--Leibman--McCutcheon (2005) showed that for any polynomial $P(x) \in \mathbb{Z}[x]$ with $P(0) = 0$, if $A \subseteq…

It is a striking and elegant fact (proved independently by Furstenberg and Sarkozy) that in any subset of the natural numbers of positive upper density there necessarily exist two distinct elements whose difference is given by a perfect…