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

Kijonka, Václav

Acta Math. Univ. Ostrav. 15, No. 1, 15-20 (2007).

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.

Reviewer: Štefan Porubský (Praha)

MSC:

11B05

Density, gaps, topology

Keywords:

asymptotic density; weighted density; strong quotient base

References:

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