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arXiv · 1701.08101

Sum-Product Type Estimates for Subsets of Finite Valuation Rings

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

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Esen Aksoy Yazici. 2017-01-27. Sum-Product Type Estimates for Subsets of Finite Valuation Rings. https://arxiv.org/abs/1701.08101

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