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Über die Dichte der Summe zweier Mengen, deren eine von positiver Dichte ist. (German) Zbl 0019.00602
Let \(A\) be a sequence of positive integers \(a\) of density \(\alpha > 0\) (\(\alpha\) is the lower bound of \(A(n)/n\) for \(n= 1, 2,\ldots\), where \(A(n)\) is the number of \(a\)’s not exceeding \(n\)]. Let \(B\) be a sequence of positive integers \(b\) containing \(1\), and let \(g(m)\) denote the least number of summands in any representation of \(m\) as a sum of \(b\)’s. Suppe that, for all \(n\), \[ \sum_{m=1}^ng(m)\leq \lambda n. \]
Let \(C\) be the sequence formed by all numbers \(a\) and \(a+b\). Landau’s modification of Erdős’s theorem (see E. Landau, Über einige neuere Fortschritte der additiven Zahlentheorie. Cambridge (1937; Zbl 0016.20201)]) asserts that the density \(\gamma\) of \(C\) satisfies
\[ \gamma\geq \alpha\left(1+\frac{1-\alpha}{2\lambda}\right). \]
This paper is devoted to a proof of the stronger inequality
\[ \gamma\geq \alpha\left(1+\frac{1-\sqrt{\alpha}}{\lambda}\right). \]
It is impossible to summarise the argument here, but the main weapon is the inequality
\[ \sum_{n=1}^{[n-\mu n]}\{A(n-m)-\alpha(n-m)\}+\tfrac 12(A(n)-\alpha n)\geq \tfrac 12 \alpha\mu n, \]
which is shown to hold for certain values of \(\mu\), including \(\mu =1-\sqrt{\alpha}\).

MSC:
11B05 Density, gaps, topology
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