Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  814.11043
Autor:  Erdös, Paul; Hall, R.R.; Tenenbaum, G.
Title:  On the densities of sets of multiples. (In English)
Source:  J. Reine Angew. Math. 454, 119-141 (1994).
Review:  Let A denote a strictly increasing sequence of integers greater than 1, and let M(A) = {ma: m \geq 1, a in A}. The authors call A a Besicovitch sequence if M(A) has an asymptotic density; if this density equals 1, then A is a Behrend sequence. It was shown by Besicovitch in 1934 that there are sequences A for which M(A) does not have a density. In 1948, Erdös obtained a criterion for A to be a Besicovitch sequence, and a short proof of his result is included in this paper.
The authors prove several theorems concerning Besicovitch sequences. For example, Theorem 3 states that A = {a1,a2,...} is a Besicovitch sequence if, for some fixed positive integer k, every gcd(ai,aj), i \ne j, has at most k distinct prime factors.
Let \tau(n,A) denote the number of divisors of n belonging to A, so M(A) = {n: \tau(n,A) > 0}, and let A(k) denote the k-th derived sequence of A, so M(A(k)) = {n: \tau(n,A) > k}. The remaining theorems provide quantitative forms of the result that \tau(n,A) ––> oo p.p. whenever A is Behrend, and these are stated in terms of the logarithmic density tk(A) of {n: \tau(n,A) \leq k}. For example, the authors prove in Theorem 5 that

inf{t0(A): |A| \leq k} = prodkj = 1 (1 - {1\over pj})

where pj denotes the j-th prime.
Reviewer:  E.J.Scourfield (Egham)
Classif.:  * 11N25 Distribution of integers with specified multiplicative constraints
                   11B75 Combinatorial number theory
Keywords:  sets of multiples; strictly increasing sequence of integers; density; Behrend sequence; Besicovitch sequences; number of divisors; logarithmic density

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