On restricted arithmetic progressions over finite fields

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 $P:mathbb{F}_p^n omathbb{F}_p^R$ of degree $d$, for $vinmathbb{F}_p^R$. We show, that under appropriate conditions, the set $A$ contains at least $c, N|S|$ arithmetic progressions of length $lleq d$ 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 $approx N^{-gamma}$.