Difference sets and power residues

Abstract

Let $p\ge 3$ be a prime and $n\ge 1$ be an integer. Let $K\subseteq {\mathbb{F}}_p$ denote a fixed subset with $0\in K$. Let $A\subseteq ({\mathbb{F}}_p{)}^n$ be an arbitrary subset such that $$\{\mathbf{a}-\mathbf{b}:\mathbf{a},\mathbf{b}\in A,\mathbf{a}\ne \mathbf{b}\}\cap K^n=\varnothing .$$ Then we prove the exponential upper bound $$|A|\le (p-|K|+1{)}^n.$$ We use in our proof the linear algebra bound method.

arXiv comment5 pages