Product of sumsets over arbitrary finite fields

Mozhgan Mirzaei, Thang Pham

Published 2018-11-19, updated 2018-11-25Version 2

Let $\mathbb{F}_q$ be an arbitrary finite field of order $q$. For $A, B, C, D\subset \mathbb{F}_q$, we show that if $|A||B||C||D|\gg q^{5/2},$ then the set $(A-B)(C-D)$ covers a positive proportion of all elements in $\mathbb{F}_q.$ Our result significantly improves the recent exponent $\frac{2}{3}-\frac{1}{13542}$ given by Murphy and Petridis.