Sum-Product Type Estimates for Subsets of Finite Valuation Rings

TL;DR

This paper establishes sum-product estimates for subsets of finite valuation rings using incidence geometry, providing bounds that relate the sizes of sum and product sets in these algebraic structures.

Contribution

It introduces new sum-product estimates for finite valuation rings based on point-plane incidence bounds, extending additive combinatorics to this setting.

Findings

Proves lower bounds for |AA+A| in finite valuation rings.

Establishes conditions under which |A^2+A^2||A+A| has a lower bound involving q and |A|.

Uses incidence geometry techniques to derive sum-product estimates.

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 $$|A^2+A^2||A+A|\gg q^{\frac{r}{2}}|A|^{\frac{3}{2}}.$$