Let \((a_n)_{n\geq 1}\) be a nondecreasing sequence of positive integer such that \[ \limsup_{n\rightarrow \infty} a_n^{1/\prod_{j=1}^{n-1} (2^j+2)}=\infty. \] Assume further that there exists \(\varepsilon>0\) such that \(a_n\geq n^{2+\varepsilon}\) for all sufficiently large \(n\). Then the number \[ \sum_{n\geq 1}\frac{1}{\sqrt{a_n}} \] is irrational.
The assumption on the sequence \((a_n)_{n\geq 1}\) is satisfied as soon as \[ \lim_{n\rightarrow \infty} a_n^{2^{-n^2/2}}=\infty, \] hence, as soon as \[ \lim_{n\rightarrow \infty} \frac{\log^2 a_n}{2^{n^2}}=\infty. \] The authors quote a number of previous results related with series \[ \sum_{n\geq 1}\frac{1}{a_n}, \] due to P. Erdős, the first author, J. Šustek, P. Rucki, T. Kanoko, T. Kurosawa, I. Shiokawa and others. The occurrence of the square root in the denominator makes a difference. The authors explain why they cannot use the Schmidt Subspace Theorem like in the work of P. Corvaja and U. Zannier [Acta Math. 193, No. 2, 175–191 (2004; Zbl 1175.11036)], nor Mahler’s method like K. Nishioka [Mahler functions and transcendence. Berlin: Springer (1996; Zbl 0876.11034)]. The proof is based on ideas of J. Liouville [“Nouvelle démonstration d’un théorème sur les irrationnelles algébriques”, C.R. Acad. Sci. Paris 18, 910–911 (1844)] and P. Erdős [J. Math. Sci. 10, 1–7 (1975; Zbl 0372.10023)]; they use a result of M. Mignotte [Ann. Fac. Sci. Toulouse, Math. (5) 1, 165–170 (1979; Zbl 0421.10022)] giving a lower bound for the difference between two algebraic numbers.