## 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.

### MSC:

 11J82 Measures of irrationality and of transcendence