Sum-product type estimates for subsets of finite valuation rings
Volume 185 / 2018
Acta Arithmetica 185 (2018), 9-18
MSC: Primary 11B30; Secondary 05C25.
DOI: 10.4064/aa170418-12-12
Published online: 1 June 2018
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 $A\subset R$, $$|AA+A|\gg \min\{q^{r}, {|A|^3}/{q^{2r-1}}\}.$$ We also show that if $|A+A|\,|A|^{2} \gt q^{3r-1}$, then $$|A^2+A^2||A+A|\gg q^{{r}/{2}}|A|^{{3}/{2}}.$$
