On restricted arithmetic progressions over finite fields

Abstract

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