Squares and difference sets in finite fields

TL;DR

This paper improves the upper bound on the size of subsets in finite fields with difference sets containing only quadratic residues, showing that for many primes the maximum size is slightly less than the trivial bound.

Contribution

It provides a refined upper bound for the maximal size of such sets in finite fields for a large class of primes, improving upon the trivial bound.

Findings

New upper bound $|B| extless \sqrt{p}$ for many primes

Bound holds for approximately three quarters of primes of form $4k+1$

Result narrows the gap between trivial and actual maximum set sizes

Abstract

For infinitely many primes $p=4k+1$ we give a slightly improved upper bound for the maximal cardinality of a set $B\subset \ZZ_p$ such that the difference set $B-B$ contains only quadratic residues. Namely, instead of the "trivial" bound $|B|\leq \sqrt{p}$ we prove $|B|\leq \sqrt{p}-1$, under suitable conditions on $p$. The new bound is valid for approximately three quarters of the primes $p=4k+1$.