Erdős’ method for determining the irrationality of products. (English) Zbl 1242.11050

The main result of the paper is the following: Let \(\varepsilon\) be a positive real number. Let \(\{a_n\}^\infty_{n=1}\) and \(\{b_n\}^\infty_{n=1}\) be two sequences of positive integers. Assume \(\{a_n\}^\infty_{n=1}\) is nondecreasing and \[ \limsup_{n\to\infty} a^{1/2^n}_n= \infty. \] Assume that for all sufficiently large \(n\) \[ n^{1+\varepsilon}\leq a_n\quad\text{and}\quad b_n\leq a^{1/\log^{1+\varepsilon}\log a_n}. \] Then the number \(x=\prod^\infty_{n=1} (1+(b_n/a_n))\) is irrational.
For the proof assume that \(x={p\over q}\), \((p, q)\in\mathbb{Z}^+\times \mathbb{Z}^+\). Than for each \((P, Q)\in\mathbb{Z}^+\times \mathbb{Z}^+\) one has \[ \Biggl| qQ\Biggl(x-{P\over Q}\Biggr)\Biggr|= |pQ- Pq| \] and now one can derive in several steps and with some ability the contradiction that there exists \[ a(P, Q)\quad\text{with}\quad 0< |pQ- Pq|< 1. \] A direct application is the following result:
Let \(\{a_n\}^\infty_{n=1}\) be an increasing sequence of positive integers such that \(\lim_{n\to\infty} a^{1/2^n}_n= \infty\).
Then the number \(\prod^\infty_{n=1} (1+ 1/a_n)^{n\to\infty}\) is irrational.
In the introduction some links are given to results connected with similar infinite products and some hints are given to some papers of Erdős.


11J72 Irrationality; linear independence over a field
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