On arithmetic progressions in symmetric sets in finite field model

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 $\mathrm{\mid }S\mathrm{\mid }=\mu \cdot q^n$ elements must contain at least $\delta(q,\mu)\cdot q^n\cdot 2^n$arithmeticprogressions$x,x+d,\ldots,x+(q-1)\cdot d$ 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 $\mathrm{\mid }S\mathrm{\mid }=\mu \cdot p^n$ 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.