Arithmetic Structure in Sparse Difference Sets

TL;DR

This paper proves that sparse difference sets in high-dimensional integer grids necessarily contain scaled copies of any given configuration, provided the set is sufficiently dense relative to its size.

Contribution

It introduces a modified argument building on Croot, Ruzsa, and Schoen's work to establish quantitative existence results for configurations in sparse difference sets.

Findings

Sparse difference sets contain scaled configurations if density exceeds a specific threshold.

The result applies to any configuration of vectors in integer lattices.

Provides a quantitative bound relating set density to the existence of configurations.

Abstract

Using a slight modification of an argument of Croot, Ruzsa and Schoen we establish a quantitative result on the existence of a dilated copy of any given configuration of integer points in sparse difference sets. More precisely, given any configuration $\{v_1,...,v_\ell\}$ of vectors in $\mathbb{Z}^d$, we show that if $A\subset[1,N]^d$ with $|A|/N^d\geq C N^{-1/\ell}$, then there necessarily exists $r\ne0$ such that $\{rv_1, ...,rv_\ell\}\subseteq A-A$.