On Arithmetic Progressions in Symmetric Sets in Finite Field Model - Hązła, Jan (The Electronic Journal of Combinatorics, 2020)
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 $|S| = μ · q^n$ elements must contain at least $δ(q, μ) · q^n · 2^n$ arithmetic progressions $x, x+d, …, x+(q-1)· d$ 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 $|S|=μ· p^n$ elements contains at least $μ^{C(p)}· p^{2n}$ 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.