Irrationality measure of sequences. (English) Zbl 1087.11049

In 1975 Erdős has given the following definition of an irrational sequence: Let \(\{a_n\}^\infty_{n=1}\) be a sequence of positive real numbers. If for every sequence \(\{c_n\}^\infty_{n=1}\) of positive integers the sum of the series \(\sum^\infty_{n=1} {1\over a_nc_n}\) is an irrational number, then the sequence \(\{a_n\}^\infty_{n=1}\) is called irrational. An example by Erdős is the sequence \(\{2^{2^n}\}^\infty_{n=1}\).
Here the authors introduce in addition to the definition of an irrational sequence a measure of irrationality of an irrational sequence \(\{a_n\}^\infty_{n=1}\) like follows: Let \({\mathfrak C}\) be the set of all sequences of positive integers, \({\mathfrak C}:= \{\{c_n\}^\infty_{n=1}\), \(c_n\in\mathbb{N}\}\). Then the number \[ \underset{\{c_n\}\in{\mathfrak C}}{}{\text{inf}}\;\lim_{q\to\infty,q\in\mathbb{N}}\,\log_q\Biggl(\min_{p\in\mathbb{N}}\,\Biggl|{1\over a_n c_n}- {p\over q}\Biggr|\Biggr) \] is called an irrationality measure of \(\{a_n\}^\infty_{n-1}\). Two criteria leading to lower bounds for irrationality measures of irrationality sequences are given. Several applications are given, for example \(\{{3^{5^n}+ n^5\over 2^{5^n}+ n^5}\}^\infty_{n=1}\) has an irrationality measure greater than or equal to \({4(\log_2 3-1)\over\log_2 3}\).
The proof consists essentially in a consideration of convergence properties of certain series.


11J82 Measures of irrationality and of transcendence