On restricted arithmetic progressions over finite fields

TL;DR

This paper proves that large subsets of finite field vector spaces contain many arithmetic progressions of bounded length with common differences in specific algebraic sets, under certain conditions.

Contribution

It establishes conditions under which subsets of finite fields contain numerous arithmetic progressions with differences in algebraic level sets, extending previous combinatorial results.

Findings

Large subsets contain many arithmetic progressions with differences in algebraic sets

Conditions are generic for sparse algebraic sets of density approximately N^{- ext{epsilon}}

Results depend on the degree of the polynomial map and the size of the subset.

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 $\approx N^{-\eps}$.