A sum-product theorem in function fields

TL;DR

This paper establishes a sum-product estimate for finite subsets in function fields and p-adic fields, showing that either the sum set or product set must be significantly larger than the original set.

Contribution

It proves a new sum-product inequality in the context of function fields and p-adic fields, extending sum-product phenomena beyond real numbers.

Findings

Sum and product sets cannot both be small in these fields.

The bound |A+A| or |AA| ≥ C|A|^{6/5 - ε} is established.

Results apply to rational function fields and p-adic 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 $\max\{|A+A|,|AA|\}\geq C |A|^{6/5-\epsilon}$. 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 $p$.