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Logarithmic density of a sequence of integers and density of its ratio set. (English) Zbl 1130.11304

The authors deal with the concepts: asymptotic density, logarithmic density and \((R)\)-density and the relations between them. A set \(A\subseteq\mathbb N\) is said to be \((R)\)-dense provided that the set \(R(A)= \{\frac ab\); \(a,b\in A\}\) is dense in \((0,+\infty)\). Note that a sufficient condition for the \((R)\)-density of a set \(A\) in terms of logarithmic densities mentioned at the beginning of the paper can be derived from the Corollary on p. 318.

MSC:

11B05 Density, gaps, topology
11B75 Other combinatorial number theory
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References:

[1] Knopp, K., Theory and Application of Infinite Series. Blackie & Son Limited, London and Glasgow, 2-nd English Edition, 1957. · Zbl 0042.29203
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[3] T, Šalát, On ratio sets of sets of natural numbers. Acta Arith.15 (1969), 173-278. · Zbl 0177.07001
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[5] Tóth, J.T., Relation between (R)-density and the lower asymptotic density. Acta Math. Constantine the Philosopher University Nitra3 (1998), 39-44.
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