An improved estimate on sums of product sets

TL;DR

This paper improves the bounds on the product set sums in finite fields, showing that fewer multiplications are needed to cover the entire field compared to previous results.

Contribution

It provides a tighter estimate, reducing the required multiple of the product set to cover the finite field from 16 to 10.

Findings

Reduced the covering number from 16 to 10 for finite field sums

Established a more efficient bound in sum-product estimates

Enhanced understanding of product set behavior in finite fields

Abstract

In a recent paper \cite{Gl} A. Glibichuk proved that if $A,B$ are subsets of an arbitrary finite filed $\F_q$, such that $|A||B|>q$, then $16AB = \F_q$. We improve this to $10AB = \F_q.$