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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 δ denote the Shnirel’man density and d L the lower asymptotic density of sets of nonnegative integers. Using Dyson’s theorem it is shown that for every h there are sets A 1 ,···,A h 0 such that

δ(C)=d L (C)=α 1 +···+α h 1

where α i =δ(A i )=d L (A i ) for i=1,···,h and C=A 1 +···+A h . This proves the general lower bounds for δ (C) and d 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 L (C)lim inf n (A 1 (n)+···+A h (n))/n=:β· Since βα 1 +···+α h the result above gives d L (C)=β proving this lower bound to be sharp all the more.

MSC:
11B05Topology etc. of sets of numbers
11B13Additive bases, including sumsets