On relations between \(f\)-density and \((R)\)-density. (English) Zbl 1235.11016

A subset \(A\) of the set \({\mathbb N}\) of positive integers, it is called a strong quotient base if for every positive rational \(p/q\) there are are infinitely many pairs \((a,b)\in A\times A\) such that \(a/b=p/q\). Let \(f\) be a non-negative real valued arithmetical function such that \(\sum_{n=1}^\infty f(n)=\infty\) and \(f(n)=o\left(\sum_{k=1}^n f(k)\right)\). The author proves that if \(f\) is in addition non-increasing and \[ \limsup_n \frac{\left(\sum_{k\leq n, k\in A} f(k)\right)}{\left(\sum_{k\leq n} f(k)\right)}=1 \] then \(A\) is a strong quotient base.


11B05 Density, gaps, topology
Full Text: EuDML Link


[1] Mišík L.: Sets of positive integers with prescribed values of densities. Math. Slovaca 52 (2002), 289-296 · Zbl 1005.11004
[2] Mišík L., Tóth J. T.: Logarithmic density of sequence of integers and density of its ratio set. Journal de Theorie des Nombers de Bordeaux 15 (2003), 309-318. · Zbl 1130.11304
[3] Strauch O., Tóth J. T.: Asymptotic density of A c N and density of the ratio set R(A). Acta Arith. 87 (1998), 67-78. · Zbl 0923.11027
[4] Strauch O., Tóth J. T.: Corrigendum to Theorem 5 of the paper ”Asymptotic density of ACN and density of the ratio set R(A)”. Acta Arith. 87 (1998), 67-78, Acta Arith. 103.2 (2002), 191-200. · Zbl 0923.11027
[5] Šalát T.: On ratio sets of sets of natural numbers. Acta Arith. 15 (1969), 273-278. · Zbl 0177.07001
[6] Šalát T.: Quotientbasen und (R)-dichte Mengen. Acta Arith. 19 (1971), 63-78. · Zbl 0218.10071
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.