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Additive properties for sets of polynomial values

Zhenchao Ge

Vol. 11 (2022), No. 3, 247–261
Abstract

We study the incrementation of the lower asymptotic density d¯(A + H) compared with d ¯(A), where H is a specific set and A is arbitrary. Ruzsa proved optimal inequalities of 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 [x] with a positive leading coefficient and H = {h(p) :  prime p}, we prove that d ¯(A + H) > d ¯(A) if and only if h(p) h(2) is not identically zero modulo any m 2.

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