Additive properties for sets of polynomial values

We study the incrementation of the lower asymptotic density $\underline{d} (A + H)$ compared with $\underline{d} (A)$ , where $H \subset ℕ$ is a specific set and $A \subset ℕ$ is arbitrary. Ruzsa proved optimal inequalities of $\underline{d} (A + H)$ for $H$ being a set of prime powers or integer powers. We generalize Ruzsa’s result to sets of polynomial values. Moreover, for $h \in ℤ [x]$ with a positive leading coefficient and $H^{'} = {h (p) : prime p}$ , we prove that $\underline{d} (A + H^{'}) > \underline{d} (A)$ if and only if $h (p) - h (2)$ is not identically zero modulo any $m \geq 2$ .

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Keywords

sumsets, essential components, circle method

Mathematical Subject Classification

Primary: 11B13, 11P55

Milestones

Received: 28 February 2022

Revised: 13 June 2022

Accepted: 29 June 2022

Published: 15 October 2022

Authors

Zhenchao Ge
Institute of Mathematical Sciences ShanghaiTech University Shanghai China

Vol. 11, No. 3, 2022

Zhenchao Ge

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors