Set avoiding squares in $\mathbb{Z}_m$

TL;DR

This paper establishes an upper bound on the size of subsets in modular arithmetic avoiding differences that are non-zero squares, extending understanding of additive combinatorics in squarefree modulus.

Contribution

The paper provides a new bound for sets avoiding square differences in rac{rac{Z}_m, specifically for squarefree m, with a bound depending on the number of prime divisors.

Findings

Bound |A| q m^{1/2}(3n)^{1.5n} for sets avoiding non-zero squares in A-A.

The result applies to all squarefree m and relates set size to prime divisors.

Advances understanding of additive structures in modular arithmetic.

Abstract

We prove that for all squarefree $m$ and any set $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero squares the bound $|A|\leq m^{1/2}(3n)^{1.5n}$ holds, where $n$ denotes the number of odd prime divisors of $m$.