OPEN
Let $\delta(m,\alpha)$ denote the density of the set of integers which are divisible by some $d\equiv 1\pmod{m}$ with $1<d<\exp(m^\alpha)$. Does there exist some $\beta\in (1,\infty)$ such that
\[\lim_{m\to \infty}\delta(m,\alpha)\]
is $0$ if $\alpha<\beta$ and $1$ if $\alpha>\beta$?
It is trivial that $\delta(m,\alpha)\to 0$ if $\alpha <1$, and Erdős could prove that the same is true for $\alpha=1$.
See also [696].
