Arithmetic and Geometric Progressions in Productsets over Finite Fields

TL;DR

This paper investigates the presence of long arithmetic and geometric progressions within productsets over finite fields, establishing conditions on set sizes and field characteristics for their guaranteed existence.

Contribution

It provides new bounds and conditions under which productsets in finite fields contain long arithmetic and geometric progressions, extending previous combinatorial results.

Findings

Productsets contain arithmetic progressions of length k ≥ 3 when k<p and set sizes are sufficiently large.

Similar results are obtained for geometric progressions in shifted productsets.

Conditions relate the size of sets and the characteristic of the finite field for progression existence.

Abstract

Given two sets $\cA, \cB \subseteq \F_q$ of elements of the finite field $\F_q$ of $q$ elements, we show that the productset $$ \cA\cB = \{ab | a \in \cA, b \in\cB\} $$ contains an arithmetic progression of length $k \ge 3$ provided that $k<p$, where $p$ is the characteristic of $\F_q$, and $# \cA # \cB \ge 3q^{2d-2/k}$. We also consider geometric progressions in a shifted productset $\cA\cB +h$, for $f \in \F_q$, and obtain a similar result.