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 denote the Shnirel’man density and the lower asymptotic density of sets of nonnegative integers. Using Dyson’s theorem it is shown that for every there are sets such that
where for and . This proves the general lower bounds for (C) and , 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 Since the result above gives proving this lower bound to be sharp all the more.