Let $T_1,T_2,....T_n$ be numbers such that $T_k= k$ no. of digits in decimal expansion of an irrational number, say $\alpha$, starting from $(\frac{k(k-1)}{2}+1)^{th}$ digit in the decimal expansion. e.g. for $\pi$, according to this definition, we have

$T_1=0.1, T_2=0.41, T_3=0.592$ and so on.

Question: Under what conditions regarding an irrational number $\alpha$ will the set of all such $T_k'$s will have a limit point of $[0,1)$ ?

$PS$: The same question for $\pi$ was asked by John Nash in one of the exams graded by him. The incident is given in the book "A beautiful mind". It was also given that this is an open problem. What I want to know is that whether there has been any work on this problem in general or in any specific cases.