On arithmetic progressions in symmetric sets in finite field model

Abstract

We consider two problems regarding arithmetic progressions in symmetric sets in the finite field (product space) model. First, we show that a symmetric set $S\subseteq {\mathbb{Z}}_q^n$ containing $|S|=\mu \cdot q^n$ elements must contain at least $\delta (q,\mu )\cdot q^n\cdot 2^n$ arithmetic progressions $x,x+d,\dots ,x+(q-1)\cdot d$ such that the difference $d$ is restricted to lie in $\{0,1{\}}^n$. Second, we show that for prime $p$ a symmetric set $S\subseteq {\mathbb{F}}_p^n$ with $|S|=\mu \cdot p^n$ elements contains at least ${\mu }^{C(p)}\cdot p^{2n}$ arithmetic progressions of length $p$. This establishes that the qualitative behavior of longer arithmetic progressions in symmetric sets is the same as for progressions of length three.