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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 $$\mu =1-\sqrt{\alpha}$$.
Show Scanned Page ##### MSC:
 11B05 Density, gaps, topology
##### Keywords:
density; sumsets; positive density
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