An additive problem for powers of primes
type: Item
instance of: scholarly article
title: An additive problem for powers of primes
zbMATH Open document ID: 0685.10035
DOI: 10.1016/0022-314X(89)90060-7
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: Let \(Q_ k\) be the set of k-th powers of nonnegative integers and \(P_ k\) the set of k-th powers of prime numbers. The question of this paper is how the density of a set increases by adding \(Q_ k\) or \(P_ k\) to it. The essentially optimal result states that \[ \underline d(A+Q_ k)\quad \geq \quad \underline d(A+P_ k)\quad \geq \quad K^{-c/\log \log K}, \] where ḏ(A) is the lower asymptotic density, \(K=1/\underline d(A)\) and c is a positive constant depending on k. The proof is based on the Hardy- Littlewood method.
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11P55
Mathematics Subject Classification ID: 11P32
zbMATH DE Number: 4122078
zbMATH Keywords: sum of sets
zbMATH Keywords: density of sets
zbMATH Keywords: powers of nonnegative integers
zbMATH Keywords: powers of prime numbers
zbMATH Keywords: lower asymptotic density
zbMATH Keywords: Hardy-Littlewood method
author: Imre Z. Ruzsa
reviewed by: Antal Balog
MaRDI profile type: Publication
cites work: Q5515488
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description: scientific article; zbMATH DE number 4122078