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 
$$\forall\epsilon>0 : |M| = \bar\eta(\epsilon) < \infty.$$

Picture below illustrates my conjecture, where I have checked this conjecture for $n < 10^6$ for each $\epsilon\in(0,1.1)$ with step 0.01
[![enter image description here][2]][2] 


The following picture based on data provided by @GottfriedHelms for $n \approx 10^{100}$ (see answer below).
[![enter image description here][3]][3]


  [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
  [2]: https://i.sstatic.net/ArpSz.png
  [3]: https://i.sstatic.net/ySVSZ.png