Let $\psi:\Bbb{N}\to\Bbb{R}_{\geq 0}$ be such that $q\mapsto q\psi(q)$ is a (weakly) decreasing function. We say a real number $x$ is $\psi$-approximable if $|qx-p|<\psi(q)$ has solutions for arbitrarily large $q\in\Bbb{N}$ with $p\in\Bbb{Z}$.
Khinchin proved that:
- Almost every $x$ is $\psi$-approximable if $\sum_q\psi(q)$ diverges.
- Almost no $x$ is $\psi$-approximable if $\sum_q\psi(q)$ converges.
My question is about the situation where we first pick an $x$ and then ask for which $\psi$ it is $\psi$-approximable.
In particular, does almost no real number $x$ have a $\psi$ such that $x$ is $\psi$-approximable and $\sum_q\psi(q)$ converges?