Sum-Product Type Estimates for Subsets of Finite Valuation Rings

Abstract

Let $R$ be a finite valuation ring of order $q^r.$ Using a point-plane incidence estimate in $R^3$, we obtain sum-product type estimates for subsets of $R$. In particular, we prove that for $A\subset R$, $$|AA+A|\gg \min \left\{q^r,\frac{|A{|}^3}{q^{2r-1}}\right\}.$$ We also show that if $|A+A||A{|}^2>q^{3r-1}$, then $$|A^2+A^2||A+A|\gg q^{\frac{r}{2}}|A{|}^{\frac{3}{2}}.$$