How do we find the smallest number of intervals covering the subset of $[0,1]$, if the intervals all have the same length of small $\epsilon$?

Question

Suppose we have a subset of $[0,1]$ (countable or uncountable) and we cover the entire subset with the smallest number of open intervals which have the same length of a small and fixed $\epsilon$. If the smallest number of intervals is $m$, what techniques can be used to solve $m$ in terms of $\epsilon$. [As a note we could approximate $m$ as function of $f$, where $m\approx f(\epsilon)$]

If there's no general techniques then what is $f$ for the following subsets of $[0,1]$?

1. $\text{Cantor Set}$
2. $\left\{{1}/{\ln(\sqrt{n})}:n\in\left[\mathbb{N}\setminus\{1\}\right]\right\}$
3. $\left\{1/n:n\in\mathbb{N}\right\}$
4. $\left\{1/\sqrt{n}:n\in\mathbb{N}\right\}$

Answer 1

First, think about how to cover finite sets of points. Given $a_1<...<a_n$, no matter what $\epsilon$ we use we should definitely make our leftmost interval $[a_1,a_1+\epsilon]$ and our rightmost interval $[a_n-\epsilon, a_n]$. Repeating this on $$\{a_1,...,a_n\}\setminus([a_1,a_1+\epsilon]\cup[a_n-\epsilon, a_n])$$ lets us recursively calculate the relevant $m$ - for a given $\epsilon$, anyways. Some care may be needed to extract a general formula for $m$ in terms of $\epsilon$.

What about infinite sets of points? Well, first note that if any cover of $A$ by finitely many closed intervals also covers all the limit points of $A$. If $A$ has only finitely many limit points $p_1,...,p_k$, then - again, for a fixed $\epsilon$ - we can try to find an optimal covering of $A$ by first thinking about how to cover the $p_i$s and then considering (for such a covering) the finitely many points in $A$ left uncovered. This applies to your last three examples: in each case there is exactly one limit point, namely $0$. In fact in these cases $0$ is also the leftmost point of the closure of $A$, which means that we definitely want our optimal covering to have as its leftmost element $[0,\epsilon]$. Note that in each case this gives us an explicit finite set of remaining points! Analyzing these may still be nontrivial, but it's a much simpler-looking problem - and in some cases is much simpler. For example, it's not hard to calculate $m$ as a function of $\epsilon$ exactly for the set $\{{1\over 2^n}: n\in\mathbb{N}\}$; this is a good exercise.

Of course these are only a couple basic observations. In general even finding the asymptotics may be quite complicated. You'll find more about this if you look at box dimension, where these values are analyzed along the way.