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 \( c N|S| \) arithmetic progressions of length \( l \leq d \) 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 \( \approx N^{-\gamma} \).