×

Sets of positive integers with prescribed values of densities. (English) Zbl 1005.11004

The author constructs a set of positive integers with prescribed lower and upper asymptotic and logarithmic densities. In particular the following result is shown:
Let \(0 \leq \alpha \leq \beta \leq \gamma \leq \delta \leq 1\) be given numbers, and let \(\underline d\), \(\overline d\) denote the lower and upper asymptotic densities. Furthermore let \(\underline \delta\), \(\overline \delta \) denote the lower and upper logarithmic densities. Then there exists a set \(A\) of positive integers with dense set of ratios such that:
\[ \underline d (A) = \alpha , \quad \underline \delta (A)=\beta , \quad \overline \delta (A) = \gamma , \quad \overline d(A)=\delta . \]

MSC:

11B05 Density, gaps, topology
PDFBibTeX XMLCite
Full Text: EuDML

References:

[1] BUKOR J.-TÓTH J. T.: On accumulation points of ratio sets of positive integers. Amer. Math. Monthly 103 (199G), 502 504. · Zbl 1050.11012
[2] GREKOS G.: Densitres in disjoint unions. Math. Slovaca 49 (1999), 255-262. · Zbl 0956.11007
[3] STRAUCH O.-TÓTH J. T.: Asymptotic density of \(A \subset N\) and density of the ratio set \(R(A)\). Acta Arith. 87 (1998), 67-78. · Zbl 0923.11027
[4] STRAUCH O.-TÓTH J. T : Distribution functions of ratio sequences. Publ. Math. Debrecen 58 (2001), 751-778. · Zbl 1183.11042
[5] ŠALÁT T.: On ratio sets of sets of natural numbers. Acta Arith 15 (1969), 273-278. · Zbl 0177.07001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.