Difference sets and positive exponential sums II: cubic residues in cyclic groups

TL;DR

This paper develops bounds on the size of subsets in cyclic groups that avoid cubic residues in their difference sets, using specially constructed exponential sums.

Contribution

It introduces a method to bound the size of difference sets avoiding cubic residues via nonnegative exponential sums in cyclic groups.

Findings

Upper bounds on the size of difference sets avoiding cubic residues

Construction of specific exponential sums for these bounds

Application to cyclic groups of various orders

Abstract

By constructing suitable nonnegative exponential sums we give upper bounds on the cardinality of any set $B_q$ in cyclic groups $\ZZ_q$ such that the difference set $B_q-B_q$ avoids cubic residues modulo $q$.