## Functiones et Approximatio Commentarii Mathematici

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

### Additive decomposability of multiplicatively defined sets

#### 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