Problem #264: Let $a_n$ be a sequence of integers such that for every...

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Progress

A possible definition of an 'irrationality sequence' (see also [262] and [263]). One example is $a_n=2^{2^n}$. In [ErGr80] they also ask whether such a sequence can have polynomial growth, but Erdős later retracted this in [Er88c], claiming 'It is not hard to show that it cannot increase slower than exponentially'.

Kovač and Tao [KoTa24] have proved that $2^n$ is not such an irrationality sequence. More generally, they prove that any strictly increasing sequence of positive integers such that $\sum\frac{1}{a_n}$ converges and\[\liminf \left(a_n^2\sum_{k>n}\frac{1}{a_k^2}\right) >0 \]is not such an irrationality sequence. In particular, any strictly increasing sequence with $\limsup a_{n+1}/a_n <\infty$ is not such an irrationality sequence.

On the other hand, Kovač and Tao do prove that for any function $F$ with $\lim F(n+1)/F(n)=\infty$ there exists such an irrationality sequence with $a_n\sim F(n)$.