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**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 = **{**a_{1},a_{2},...**}** is a Besicovitch sequence if, for some fixed positive integer k, every gcd(a_{i},a_{j}), 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 t_{k}(A) of **{**n: \tau(n,A) \leq k**}**. For example, the authors prove in Theorem 5 that **inf****{**t_{0}(A): |A| \leq k**}** = **prod**^{k}_{j = 1} **(**1 - {1\over p_{j}}**)** where p_{j} 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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