A sum-product theorem in function fields

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 $ε>0$ there exists a constant $C$ dependent only on $ε$ and $q$ such that $\max\{|A+A|,|AA|\}\geq C |A|^{6/5-ε}$. 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 $p$.

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