## 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.

### MSC:

 11B05 Density, gaps, topology

### Keywords:

asymptotic density; weighted density; strong quotient base
Full Text:

### References:

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