I was inspired by this topic in Math.SE.

Conjecture

Let $M$ be a set of all $n$ such that $$H_n - \lfloor{H_n\rfloor} < \frac{1}{n^{1+\epsilon}}.$$ Then $$\forall\epsilon>0 : |M| = \bar\eta(\epsilon) < \infty.$$

Also I suspect that $\bar\eta$ is a decreasing function on $(0,1]$. Please, let me know if it is obviously (or not obviously) false.