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Best possible results on the density of sumsets. (English) Zbl 0722.11007
Analytic number theory, Proc. Conf. in Honor of Paul T. Bateman, Urbana/IL (USA) 1989, Prog. Math. 85, 395-403 (1990).
[For the entire collection see Zbl 0711.00008.] Let $\delta$ denote the Shnirel’man density and $d\sb L$ the lower asymptotic density of sets of nonnegative integers. Using Dyson’s theorem it is shown that for every $h\in {\bbfN}$ there are sets $A\sb 1,...,A\sb h\subseteq {\bbfN}\sb 0$ such that $$ \delta (C)=d\sb L(C)=\alpha\sb 1+...+\alpha\sb h\le 1 $$ where $\alpha\sb i=\delta (A\sb i)=d\sb L(A\sb i)$ for $i=1,...,h$ and $C=A\sb 1+...+A\sb h$. This proves the general lower bounds for $\delta$ (C) and $d\sb L(C)$, which are given in Mann’s theorem and in the first case of Kneser’s theorem respectively, to be the best possible ones. It should be mentioned that Kneser even obtained $d\sb L(C)\ge \liminf\sb{n\to \infty}(A\sb 1(n)+...+A\sb h(n))/n=:\beta.$ Since $\beta \ge \alpha\sb 1+...+\alpha\sb h$ the result above gives $d\sb L(C)=\beta$ proving this lower bound to be sharp all the more.
11B05Topology etc. of sets of numbers
11B13Additive bases, including sumsets