## Asymptotic density of $$A\subset\mathbb N$$ and density of the ratio set $$R(A)$$.(English)Zbl 0923.11027

Acta Arith. 87, No. 1, 67-78 (1998); correction ibid. 103, No. 2, 191-200 (2002).
For $$A\subseteq\mathbb N$$ denote by $$\underline d(A)$$, $$\bar d(A)$$ and $$d(A)$$ the lower, upper and asymptotic density of $$A$$, respectively. The reviewer proved [cf. T. Šalát, Acta Arith. 15, 273–278 (1969; Zbl 0177.07001); Corrigendum, ibid. 16, 103 (1969)] that the set $R(A)=\left\{{a\over b}:a,b\in A\right\}$ is a dense subset of $$\mathbb R^+= (0, +\infty)$$ provided that $$d(A)>0$$ or $$\bar d(A)=1$$ and that $$\underline d(A) >0$$ is not sufficient for density of $$R(A)$$ in $$\mathbb R^+$$. The authors prove further results concerning sets $$R(A)$$ and prove among other results that if $$\underline d(A)\geq{1\over 2}$$, then $$R(A)$$ is dense in $$\mathbb R^+$$ and for each $$t\in [0,{1\over 2})$$ there is a set $$A\subseteq \mathbb N$$ with $$\underline d(A)=t$$ such that $$R(A)$$ is not dense in $$\mathbb R^+$$.
The corrigendum states that theorem 5 is wrong. It is given a new form.

### MSC:

 11B05 Density, gaps, topology 11K06 General theory of distribution modulo $$1$$ 11K31 Special sequences

### Keywords:

ratio sets; asymptotic density

Zbl 0177.07001
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