A sum-product theorem in function fields (Q2929548)

Let \(A\) be a finite subset of \(\mathbb F_q((t^{-1}))\), the field of formal Laurent series in \(1/t\) over a finite field \(\mathbb F_q\). Denote by \(|A|\) the cardinality of \(A\). The authors study the sumset \(A+A=\{ a+b:\;a,b\in A\}\) and the product set \(AA=\{ ab:\;a,b\in A\}\). They prove that, for any \(\varepsilon >0\), NEWLINE\[NEWLINE \max \{|A+A|,|AA|\} \geq C|A|^{\frac65-\varepsilon} NEWLINE\]NEWLINE where \(C\) depends only on \(\varepsilon\) and \(q\). A similar result is announced (without a full proof) for subsets of \(\mathbb Q_p\).