On restricted arithmetic progressions over finite fields

Abstract

Let A be a subset of $\F_p^n$, the $n$-dimensional linear space over the prime field $\F_p$ of size at least $\de N$ $(N=p^n)$, and let $S_v=P^{-1}(v)$ be the level set of a homogeneous polynomial map $P:\F_p^n\to\F_p^R$ of degree $d$, and $v\in\F_p^R$. We show, that under appropriate conditions, the set $A$ contains at least $c\, N|S|$ arithmetic progressions of length $l\leq d$ with common difference in $S_v$, where c is a positive constant depending on $\de$, $l$ and $P$. We also show that the conditions are generic for a class of sparse algebraic sets of density $\approx N^{-\eps}$.