A note on sum-product estimates over finite valuation rings

Abstract

Let $\mathcal{R}$ be a finite valuation ring of order $q^r$ with $q$ a power of an odd prime number, and $\mathcal{A}$ be a set in $\mathcal{R}$. In this paper, we improve a recent result due to Yazici (2018) on a sum-product type problem. More precisely, we will prove that 1. If $|\mathcal{A}|\gg q^{r-\frac{1}{3}}$, then $$\max \left\{|\mathcal{A}+\mathcal{A}|,|{\mathcal{A}}^2+{\mathcal{A}}^2|\right\}\gg q^{\frac{r}{2}}|\mathcal{A}{|}^{\frac{1}{2}}.$$ 2. If $q^{r-\frac{3}{8}}\ll |\mathcal{A}|\ll q^{r-\frac{1}{3}}$, then $$\max \left\{|\mathcal{A}+\mathcal{A}|,|{\mathcal{A}}^2+{\mathcal{A}}^2|\right\}\gg \frac{|\mathcal{A}{|}^2}{q^{\frac{2r-1}{2}}}.$$ 3. If $|\mathcal{A}+\mathcal{A}||\mathcal{A}{|}^2\gg q^{3r-1}$ and $2q^{r-1}\le |\mathcal{A}|\ll q^{r-\frac{3}{8}}$, then $$\max \left\{|\mathcal{A}+\mathcal{A}|,|{\mathcal{A}}^2+{\mathcal{A}}^2|\right\}\gg q^{r/3}|\mathcal{A}{|}^{2/3}.$$