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January 2006 Additive decomposability of multiplicatively defined sets
Christian Elsholtz
Funct. Approx. Comment. Math. 35: 61-77 (January 2006). DOI: 10.7169/facm/1229442617

Abstract

Let ${\mathcal{Q}}({\mathcal{T}})$ denote the set of integers which are composed of prime factors from a given set of primes ${\mathcal{T}}$ only. Suppose that $\mathcal{A}+\mathcal{B} \subseteq \mathcal{Q}'(\mathcal{T})$, where $\mathcal{Q}(\mathcal{T})$ and $\mathcal{Q}'(\mathcal{T})$ differ at finitely many elements only. Also assume that $\sum_{p \leq x, p \in \mathcal{T}} \frac{\log p}{p} = \tau \log x + O(1)$. We prove that $\mathcal{A}(N) \mathcal{B}(N) = O(N(\log N)^{2 \tau})$ holds. In the case $\tau \geq \frac{1}{2}$ we give an example where both $\mathcal{A}(N)$ and $\mathcal{B}(N)$ are of order of magnitude $\frac{N^\frac{1}{2}}{(\log N)^{\frac{1}{4}}}$, which shows that this is close to best possible.

Citation

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Christian Elsholtz. "Additive decomposability of multiplicatively defined sets." Funct. Approx. Comment. Math. 35 61 - 77, January 2006. https://doi.org/10.7169/facm/1229442617

Information

Published: January 2006
First available in Project Euclid: 16 December 2008

zbMATH: 1196.11139
MathSciNet: MR2271607
Digital Object Identifier: 10.7169/facm/1229442617

Subjects:
Primary: 11P32
Secondary: 11E25, 11N36

Rights: Copyright © 2006 Adam Mickiewicz University

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