Let $\delta>0$ and define $N(\delta)$ to be the least number of closed intervals of length $\delta$ necessary to cover the set $S:=\{0,1,1/2,1/3,\dots\}$. For example, it is clear that $N(1/2)=2$.
I am interested in the problem of finding precise asymptotics for $N(\delta)$ as $\delta\to 0$. I encountered this problem when calculating the box dimension of the set $S$ in Kenneth Falconer's book "Fractal geometry" (Example 2.7, page 37). There, it is shown that $k\leq N(\delta)\leq 2k$ where $k$ is the integer satisfying $\frac{1}{(k-1)k}>\delta\geq \frac{1}{k(k+1)}$, see Prove that the box dimension of $\{0,1,\frac{1}{2},\frac{1}{3},...\} $is$ \frac{1}{2}$. If I didn't make any mistakes, $k$ is given by $$ k=\left\lceil\frac{-1+\sqrt{1+4/\delta}}{2}\right\rceil. $$
I am tempted to think that in fact there exists a constant $C$ such that $N(\delta)=C\sqrt{1/\delta}+O(1)$ as $\delta\to 0$. If this is true, I would like to see a proof of this fact and know if $C$ has a nice expression.
I can prove that the best strategy to cover $S$ with intervals of length $\delta$ is a greedy one, in the following sense. First, you take the interval $[0,\delta]$, then you find the largest integer $m$ so that $1/m\not\in [0,\delta]$ and consider the interval $[1/m, 1/m+\delta]$, and keep repeating this until you exhaust all the elements of $S$. To see why this is true, note that $0$ must be on some interval of length $\delta$. Obviously, $[0,\delta]$ is the best choice (since there are no elements to the "left" of zero). Next, $1/m$ (where $m$ is as above) must be in a different interval from $0$, and we can also assume that it in fact starts at $1/m$. And so on.
Let $a_0=0<a_1<a_2<\cdots$ be the starting points of the intervals in this "greedy configuration". There is a recurrence relation $$ a_{k+1}=\frac{1}{\left\lceil{\dfrac{1}{a_k+\delta}}\right\rceil-1} $$ and $N(\delta)$ is then the smallest $k$ so that $a_k=\infty$. I wrote a simple program in sage for calculating $N(1/n)$, $n=2,3,\dots$, and when dividing by $\sqrt{n}$ the value seems to be around $1.86002325...$ for $n$ around $100.000.000.000$ (for $n$ around say $10.000$ the first two decimal digits aren't steady), but don't take this too seriously.
The first few values of $N(1/n)$, $n=2,3,\dots$, are 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11 ,... which matches OEIS A073047 for the first few values, but for $n=57$ I get $N(1/57)=14$, while in this OEIS sequence the corresponding value is $13$. In general, the elements of my sequence tend to be somewhat larger than those of the OEIS sequence, and I could not find anything else in the OEIS.
