Additive properties of product sets in an arbitrary finite field

TL;DR

This paper proves that for large enough subsets in any finite field, certain scaled product sets cover the entire field, revealing new additive properties of product sets in finite fields.

Contribution

It establishes new bounds on product set sizes that guarantee coverage of the entire finite field, generalizing previous results to arbitrary finite fields.

Findings

For subsets A, B with |A||B|>q, 16AB=Fq.

For subsets X, Y with |X||Y|≥2q, 8XY=Fq.

These results extend additive combinatorics in finite fields.

Abstract

It is proved that for any two subsets $A$ and $B$ of an arbitrary finite field $\Fq$ such that $|A||B|>q$ the identity $16AB=\Fq$ holds. Moreover, it is established that for every subsets $X, Y\subset \Fq$ with the property $|X||Y|\geqslant 2q$ the equality $8XY=\Fq$ holds.