Difference sets and the slice rank bounding method

Gábor Hegedűs

Published 2018-01-19Version 1

Let $p\geq 3$ be a prime, $k\geq 2$ and $n\geq 1$ be integers. Denote by $Q(k)$ the set of $k$th power residues modulo $p$. Suppose that $1<gcd(k,p-1)<p-1$. Let $A\subseteq ({\mathbb Z _p})^n$ be an arbitrary subset such that $$ \{ \mathbf{a}-\mathbf{b}:~\mathbf{a},\mathbf{b}\in A,\mathbf{a}\neq \mathbf{b}\}\cap (Q(k))^n=\emptyset. $$ Then we prove the exponential upper bound $$ |A|\leq 2\Big( \frac{(p-1)(d-1)}{d} + 1 \Big)^n, $$ where $d=gcd(k,p-1)$. We use in our proof Tao's slice rank bounding method.