Asymptotic distance and its application (Q627379)

For a set \(A\subseteq\mathbb N\), define the asymptotic density of \(A\) \[ d(A)= \lim_{n\to\infty} {|A\cap\{1, 2,\dots, n\}|\over n} \] (provided this limit does exist). For two infinite disjoint sets \(A,B\subseteq\mathbb N\), let \[ D_1= \varliminf_{x\in A,y\in B,x< y}\Biggl({y\over x}- 1\Biggr), \] and define the asymptotic distance of the sets \(A\) and \(B\), \[ D(A,B)= \min(D_1(A, B), D_1(B, A)). \] In the present paper the authors investigate some properties of these concepts, and as an application of the concept of asymptotic distance, give a new criterion for an infinite series to be a Liouville number as follows: Let \(\{a_n\}^\infty_{n=1}\) be a nondecreasing sequence of positive integers with \(\gamma= \varliminf_{n\to\infty} n^{-1}\log_2\log_2 a_n> 0\). If there are numbers \(\alpha> \beta\geq\gamma\) such that \(D(A, B)= 0\), where the sets \[ A= \{n\in\mathbb N; n^{-1}\log_2\log_2 a_n> \alpha\},\;B= \{n\in\mathbb N; n^{-1}\log_2\log_2 a_n< \beta\}, \] then the sum of the series \(\sum^\infty_{n=1} 1/a_n\) is a Liouville number. By this criterion the authors give some new examples of Liouville numbers. Moreover, an open problem is also proposed.