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Let $S_n:=1+\frac12+\frac13+\ldots+\frac1n$. Is it true that the set of $n\in\mathbb N$ such that $$S_n-[S_n]<\dfrac{1}{n^2}$$ is infinite?
Here, $[x]$ represents the largest integer not exceeding $x$.

This question has been asked previously on math.SE without receiving any answers.

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  • $\begingroup$ Who is $H_n$? what's its relation to the question? The second part is very vague as it is, even when I try to compare this post to the math.se question. Can you re-edit? $\endgroup$
    – Amir Sagiv
    Commented Dec 27, 2016 at 7:54
  • $\begingroup$ BTW I should say that I appreciate that you have added link to your post on math.SE. (There are many users who repost question here without mentioning that it has been previously posted on another site.) $\endgroup$
    – Martin Sleziak
    Commented Jan 15, 2017 at 12:21
  • $\begingroup$ Any background? Any numerical data? I am just curious--the question seems interesting to me as it is. One may ask about the opposite inequality $\ldots\ > -\frac 1{n^2}\ $ as well. $\endgroup$
    – Włodzimierz Holsztyński
    Commented Jan 19, 2017 at 0:50
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    $\begingroup$ The heuristic in the post on MSE is natural to use, but you need to be more careful -- I'm pretty sure $H_n$ is not equidistributed mod 1. Indeed, it has the same rate of growth as $\log n$, and growth rate arguments (e.g. math.stackexchange.com/questions/1950100/…) suffice to show that $\log n$ is not equidistributed mod 1. The comments in that post might help with this problem. $\endgroup$
    – Kevin Casto
    Commented Jan 19, 2017 at 4:38

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