Product Sets of Arithmetic Progressions in Function Fields

TL;DR

This paper investigates the size of product sets formed by finite arithmetic progressions of polynomials over finite fields, providing bounds and applications to function field analogs of Erdős' multiplication table problem.

Contribution

It establishes a uniform lower bound for product set sizes of polynomial progressions and applies these results to solve function field versions of Erdős' multiplication table problem.

Findings

Derived a lower bound for product set sizes in polynomial arithmetic progressions

Applied results to solve function field variants of Erdős' multiplication table problem

Provided insights into the structure of polynomial product sets over finite fields

Abstract

We study product sets of finite arithmetic progressions of polynomials over a finite field. We prove a lower bound for the size of the product set, uniform in a wide range of parameters. We apply our results to resolve the function field variants of Erd\H{o}s' multiplication table problem.