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On the densities of sets of multiples. (English) Zbl 0814.11043

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, Erdős 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 \(A = \{a_1,a_2,\dots\}\) is a Besicovitch sequence if, for some fixed positive integer \(k\), every \(\text{gcd}(a_i,a_j)\), \(i \neq j\), 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 \(\tau(n,A) \to \infty\) 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 \[ \inf\{t_0(A): |A| \leq k\} = \prod^k_{j = 1} \left(1 - {1\over p_j}\right) \] where \(p_j\) denotes the \(j\)-th prime.

MSC:

11N25 Distribution of integers with specified multiplicative constraints
11B75 Other combinatorial number theory
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