A sum-product theorem in function fields

Abstract

Let $A$ be a finite subset of $\ffield$, the field of Laurent series in $1/t$ over a finite field ${\mathbb{F}}_q$. We show that for any $\epsilon >0$ there exists a constant $C$ dependent only on $\epsilon$ and $q$ such that $\max \{|A+A|,|AA|\}\ge C|A{|}^{6/5-\epsilon }$. In particular such a result is obtained for the rational function field ${\mathbb{F}}_q(t)$. Identical results are also obtained for finite subsets of the $p$-adic field ${\mathbb{Q}}_p$ for any prime $p$.

Simplification of argument and note that methods also work for the p-adics