I was inspired by [this][1] topic on Math.SE.  
Suppose that $H_n = \sum\limits_{k=1}^n \frac{1}{k}$ - $n$th harmonic number. Then
## Conjecture ##
> Let $M$ be a set of all $n$ such that 
$$H_n - \lfloor{H_n\rfloor} < \frac{1}{n^{1+\epsilon}}.$$
Then 
$$\forall\epsilon>0 : |M| = \bar\eta(\epsilon) < \infty.$$

Please, let me know if it is obviously (or not obviously) false. 

  [1]: http://math.stackexchange.com/questions/2062960/there-exist-infinite-many-n-in-mathbbn-such-that-s-n-s-n-frac1n2?noredirect=1#comment4336226_2062960