Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 409.10043
Autor: Erdös, Paul; Saffari, B.; Vaughan, R.C.
Title: On the asymptotic density of sets of integers. II. (In English)
Source: J. London Math. Soc., II. Ser. 19, 17-20 (1979).
Review: [Part I, cf. ibid. 13, 475-485 (1976; Zbl 333.10039)]
Let A and B be a pair of direct factors of N^*, the set of positive integers; that is a pair of subsets A and B of N^* such that every n in N^* can be written uniquely as n = a· b, with a in A and b in B. Let S\subset N^* and d(S) denote the asymptotic density of S whenever it exists.Let H(S) = sumn in s 1/n . It has been shown by Saffari that in the convergent case, the sets A and B habe asymptotic densities: d(A) = \frac 1{H(B)} and d(B) = \frac 1{H(A)}. In this paper the authors settle (in the affirmative) the first two open problems stated by Saffari. In fact they prove: Theorem 1. The direct factors A and B have asymptotic densities in the divergent case H(A) = H(B) = oo and d(A) = 0. Theorem 2. In the divergent case H(A) = H(B) = oo, we have sumb in A 1/b = sumb in B 1/p = oo.
Reviewer: D.Suryanarayana
Classif.: * 11B83 Special sequences of integers and polynomials
Keywords: asymptotic density; sets of integers; direct factors
Citations: Zbl.333.10039
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag