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A note on sum-product estimates over finite valuation rings

Volume 198 / 2021

Acta Arithmetica 198 (2021), 187-194 MSC: Primary 11B30; Secondary 05C25. DOI: 10.4064/aa200414-20-10 Published online: 29 January 2021

Abstract

Let $\mathcal R$ be a finite valuation ring of order $q^r$ with $q$ a power of an odd prime number, and let $\mathcal A \subset \mathcal R$. We improve a recent result due to Yazici (2018) on a sum-product type problem. More precisely, we prove that
$\bullet$ if $|\mathcal A |\gg q^{r- {1}/{3}}$, then $\max \lbrace |\mathcal A +\mathcal A |, |\mathcal A ^2+\mathcal A ^2| \rbrace \gg q^{ {r}/{2}}|\mathcal A |^{ {1}/{2}};$
$\bullet$ if $q^{r- {3}/{8}}\ll |\mathcal A |\ll q^{r- {1}/{3}}$, then $\max \lbrace |\mathcal A +\mathcal A |, |\mathcal A ^2+\mathcal A ^2| \rbrace \gg \frac {|\mathcal A |^2}{q^{ {(2r-1)}/{2}}};$
$\bullet$ if $|\mathcal A +\mathcal A |\,|\mathcal A |^2\gg q^{3r-1}$ and $2q^{r-1}\le |\mathcal A |\ll q^{r- {3}/{8}}$, then $\max \lbrace |\mathcal A +\mathcal A |, |\mathcal A ^2+\mathcal A ^2| \rbrace \gg q^{r/3}|\mathcal A |^{2/3}.$

Authors

• Duc Hiep PhamUniversity of Education
Vietnam National University, Hanoi
144 Xuan Thuy, Cau Giay, Hanoi, Vietnam
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