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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

Citations:

Zbl 0177.07001
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