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Logarithmic density and measures on semigroups. (English) Zbl 0860.11005

A set \(A\subseteq\mathbb{N}\) is said to be a multiplicative ideal provided that together with \(a\in A\), it contains also the number \(an\) \((n\in\mathbb{N})\). It is well-known that every multiplicative ideal has logarithmic density [cf. H. Davenport and P. Erdös, Acta Arith. 2, 147-151 (1936; Zbl 0015.10001)]. The author has extended this result to structures (so called quasi-ideals) \(A\) such that from \(a\), \(b\in A\), \(an\in A\), it follows that \(bn\in A\) [cf. the author, Acta. Arith. 42, 79-90 (1982; Zbl 0495.10034)]. In this paper a new proof of this result is given. The proof is based on a result on convolution of measures on discrete semigroups.

MSC:

11B05 Density, gaps, topology
43A10 Measure algebras on groups, semigroups, etc.
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References:

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