Quantitative twisted patterns in positive density subsets

Quantitative twisted patterns in positive density subsets, Discrete Analysis 2024:1, 17 pp.

A major theme of arithmetic combinatorics is the structures that can be found inside difference sets of dense sets of integers, or dense subsets of more general groups. Many results in this direction concern subsets of finite groups, but there are also interesting results about infinite groups such as $\mathbb Z^d$, with an appropriately chosen notion of density.

One such result, which is in a similar spirit to the results of this paper, concerns the question of what distances can be found in difference sets of dense sets. To be more precise, let $A$ be a subset of $\mathbb Z^d$ with positive upper Banach density $d^*(A)$ (defined to be the lim sup of $N^{-d}|A\cap[0,N)^d$). Then what can we say about the set $D(A)$ of all positive integers of the form $\|x-y\|_2^2$ with $x,y\in A$?

If $d=1$, then $D(A)$ is the set of squares of elements of $A-A$, so the question is not asking anything more interesting than “What can we say about $A-A$?” In general, the elements of $D(A)$ are sums of $d$ squares, so for $d\leq 3$ we have a restriction on the integers that can appear in $D(A)$, whereas by Lagrange’s theorem we have no such restriction for $d\geq 4$.

Another obvious remark is that if all the coordinates of all the points in $A$ are divisible by $m$, then so are all the elements of $D(A)$. When $d\geq 5$, a theorem of Magyar shows that this is almost the only restriction. It says that for every $d\geq 5$ and every $\epsilon>0$ there exists a positive integer $k$ such that for every set $A\subset\mathbb Z^d$ of upper Banach density at least $\epsilon$ the set $D(A)$ contains all sufficiently large multiples of $k$. Note that the “sufficiently large” depends on $A$ but $k$ depends only on the density of $A$.

In the light of this result, it is tempting to ask similar results about the images of distance sets under other functions, and in particular other polynomials. That is the theme of this paper.

In fact, it was also the theme of some earlier papers, where results of the above type were proved, but with one important difference: the $k$ that was obtained in those results depended on the set $A$. This paper aims to remedy that defect by obtaining uniform versions of the results, so that they match better the result of Magyar.

A wide class of polynomials was defined in a paper of Björklund and the second author, and a non-uniform Magyar-type result was proved for all the polynomials in that class. In this paper that result is upgraded to a uniform one. Two examples that the authors focus on are the polynomials $x^2+y^2-z^2$ and $xy-z^2$. If $F$ is one or other of these two polynomials, then for every $\epsilon>0$ there exists $k$ such that for every set $A\subset\mathbb Z^3$ of upper Banach density at least $\epsilon$ we have that $k\mathbb Z\subset F(A-A)$.

The polynomial $xy-z^2$ is the determinant of the matrix $\begin{pmatrix}z&-x\\ y&-z\\ \end{pmatrix}$, and the set of all such matrices forms an additive subgroup of the group of all $2\times 2$ integer matrices of trace 0. The group $SL_2(\mathbb Z)$ acts on this group by conjugation, which preserves the determinant, and the set of matrices just defined is an orbit of the action. The wider class of polynomials comes from generalizing this observation.

In this paper, the authors show that under certain additional conditions, one can prove a uniform version of this result. In particular, they obtain uniform statements for the polynomials $x^2+y^2-z^2$ and $xy-z^2$.

One of the main tools in the proof is a generalization of the Furstenberg-Sárközy theorem that applies to certain polynomials that do not necessarily have a zero constant term.

An appealing corollary of the results in this paper is that if $R$ is an integer polynomial of degree at least 2 with zero constant term and we set $P(x,y)$ to be $x+R(y)$, then for every $\epsilon>0$ there exists $k$, depending only on $\epsilon$ and $R$, such that if $A\subset\mathbb Z^2$ is any set of upper Banach density at least $\epsilon$, then $k\mathbb Z\subset P(A-A)$.