On some multiplicative properties of large difference sets

TL;DR

This paper investigates the multiplicative structure of difference sets in modular integers, providing quantitative insights into their product sets and showing the smallness of their multiplicative covering numbers.

Contribution

It offers a quantitative version of Fish's structural result and demonstrates that the multiplicative covering number of difference sets is always small.

Findings

Quantitative structure of product sets (A-A)(A-A) in composite moduli.

Small multiplicative covering number for any difference set.

Extension of Fish's results to composite moduli.

Abstract

In our paper we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq \mathbb{Z}/q\mathbb{Z}$ in the case of composite $q$. We obtain a quantitative version of a result of A. Fish about the structure of the product sets $(A-A)(A-A)$. Also, we show that the multiplicative covering number of any difference set is always small.