Functiones et Approximatio Commentarii Mathematici

Additive decomposability of multiplicatively defined sets

Christian Elsholtz

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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.

Article information

Source
Funct. Approx. Comment. Math., Volume 35 (2006), 61-77.

Dates
First available in Project Euclid: 16 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.facm/1229442617

Digital Object Identifier
doi:10.7169/facm/1229442617

Mathematical Reviews number (MathSciNet)
MR2271607

Zentralblatt MATH identifier
1196.11139

Subjects
Primary: 11P32: Goldbach-type theorems; other additive questions involving primes
Secondary: 11N36: Applications of sieve methods 11E25: Sums of squares and representations by other particular quadratic forms

Keywords
inverse Goldbach problem additive decompositions of sets sums of two squares

Citation

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


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