Sets whose differences avoid squares modulo m

TL;DR

This paper investigates the size of subsets of integers modulo m whose differences avoid quadratic residues, establishing bounds that depend on how slowly a function epsilon(m) approaches zero.

Contribution

It proves that for almost all m, subsets avoiding non-zero quadratic residues in their differences are significantly smaller than m, with bounds depending on epsilon(m).

Findings

For almost all m, such subsets have size at most m^{1/2 - epsilon(m)}.

The bound holds for any epsilon(m) tending to zero arbitrarily slowly.

The result links the structure of difference sets to quadratic residues modulo m.

Abstract

We prove that if $\varepsilon(m)\to 0$ arbitrarily slowly, then for almost all $m$ and any $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero quadratic residues we have $|A|\leq m^{1/2-\varepsilon(m)}.$