Divisibility problems for function fields

TL;DR

This paper explores divisibility properties of sum sets and shifted products in function fields, achieving improved results over integer cases by utilizing a novel large sieve for sparse moduli.

Contribution

It introduces enhanced bounds for divisibility problems in function fields, extending and improving upon classical integer results using a new large sieve technique.

Findings

Better bounds for divisibility properties in function fields

Application of a new large sieve for sparse sets of moduli

Improved understanding of sum set and shifted product divisibility in function fields

Abstract

We investigate three combinatorial problems considered by Erd\"os, Rivat, Sark\"ozy and Sch\"on regarding divisibility properties of sum sets and sets of shifted products of integers in the context of function fields. Our results in this function field setting are better than those previously obtained for subsets of the integers. These improvements depend on a version of the large sieve for sparse sets of moduli developed recently by the first and third-named authors.