Difference sets and power residues

TL;DR

This paper establishes an exponential upper bound on the size of subsets in finite fields avoiding certain difference sets related to power residues, using linear algebra techniques.

Contribution

It provides a new exponential upper bound for subsets avoiding specific difference sets in finite fields, employing the linear algebra bound method.

Findings

Proved an exponential upper bound: |A| ≤ (p - |K| + 1)^n.

Applied linear algebra bound method to difference set problems.

Extended understanding of difference sets and power residues in finite fields.

Abstract

Let $p\geq 3$ be a prime and $n\geq 1$ 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 $$\{ \mathbf{a}-\mathbf{b}:~\mathbf{a},\mathbf{b}\in A,\mathbf{a}\neq \mathbf{b}\}\cap K^n=\emptyset. $$ Then we prove the exponential upper bound $$ |A|\leq ( p-|K|+ 1 )^n. $$ We use in our proof the linear algebra bound method.