One possible definition of an 'irrationality sequence' (see also [262] and [264]). A folklore result states that $\sum \frac{1}{a_n}$ is irrational whenever $\lim a_n^{1/2^n}=\infty$.

Kovač and Tao [KoTa24] have proved that any strictly increasing sequence such that $\sum \frac{1}{a_n}$ converges and $\lim a_{n+1}/a_n^2=0$ is not such an irrationality sequence. On the other hand, if\[\liminf \frac{a_{n+1}}{a_n^{2+\epsilon}}>0\]for some $\epsilon>0$ then the above folklore result implies that $a_n$ is such an irrationality sequence.

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This page was last edited 28 September 2025.

1 comment on this problem

• Junnosuke Koizumi showed some "endpoint" results in the other direction, such as that $a_n=\lfloor \alpha^{2^n}\rfloor$ is an irrationality sequence of this type for all but countably many $\alpha>1$.

  Vjeko_Kovac — 16:15 on 11 Aug 2025

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