@Wojowu: I've hopefully answered your question in the OP.

@MartinSleziak. I'm not so convinced that would make the proof shorter, since you first need to show that ${\sf d}^\ast(X)=\limsup_n\frac{|X \cap [\![1, b_n]\!]|}{b_n}$. AFAICS, this can't avoid a rough estimate of the counting function of $X$, which is possible, e.g., by letting $N$ be as in Anthony Quas' comment and considering that $\frac{x}{y} \le \frac{x+z}{y+z}$ for all $x,y,z \in \mathbf R^+$ with $x \le y$. In any case, thank you for your comment!

Because "about half a page" is not "half a page" (but an approximation in excess, which takes into account 5 lines for the statement and two more lines for a second point, which I didn't mention in the OP), and because we were working with $b_n \le N < b_{n+1}$ rather than $a_n<N \le a_{n+1}$, which is, indeed, much better (thank you!). Anyway, we would still prefer a reference, if any.

You have both $a$ and $\alpha$, and may want to fix it. Also, have your read the comments to Geoff Robinson's answer here: mathoverflow.net/a/206941/16537?

Btw, I've just realized a mistake in one of my comments: Strauch & Tóth's paper doesn't look for the minimal lower asymptotic density of a set $X\subseteq{\bf N}^+$ s.t. the ratio set of $X$ is dense in $[0,\infty[$; the answer to this question is $0$, by taking $X$ to be the set of all positive rational primes and having a look at the bottom of p. 155 in W. Sierpiński, Elementary Theory of Numbers, PWN, Warszawa, 1964. What Strauch & Tóth prove is, instead, that $\frac{1}{2}$ is the minimal $\gamma\ge 0$ s.t. the ratio set of $X$ is dense in $[0,\infty[$ whenever ${\sf d}_\ast(X)\ge\gamma$.

However, this doesn't mean much to me, when compared with the length of the proof of the main theorem (viz., Th. 3 on p. 77) in Luca & Porubský's 2005 paper, since the latter is, to my eyes, much stronger (and more constructive) than Mišík's Th. 2. Again, I've not read Mišík's paper carefully (I should, but had no time). Yet, I mentioned your comment to other guys who did it (I believe!), and they replied that they aren't aware of any issue with Mišík's paper. So would you mind to be more specific on what you think is missing in Mišík's proof? I'm very interested, and other peps may be too.

@KevinO'Bryant: Mišík's paper has two main results, Th. 1 (p. 291) & Th. 2 (p. 293): Th. 1 is about the independence of the lower and upper $f$-densities, ${\underline{d}}_f$ and ${\overline{d}}^f$ in Mišík's notation, associated with a function $f:{\bf N}^+\to{\bf R}^+$ s.t. $\sum_{n\ge 1}f(n)=\infty$ and $f(n)/\sum_{i=1}^nf(i)\to 0$ as $n\to\infty$, while Th. 2 is about the independence of the upper and lower asymptotic and logarithmic densities. In fact, the proof of Th. 2 is just 2+1/2 pages, since we don't really need Lemma 1 here (even if this is not mentioned in the paper...). [tbc]

@KevinO'Bryant: I join Charles in his request. What are you alluding to? I don't claim to have read Mišík's paper carefully, but as far as I can say, there is no problem in there (incidentally, the paper is cited, e.g., in: F. Luca, C. Pomerance, and Š. Porubský, Sets with prescribed arithmetic densities, Unif. Distr. Theory 3 (2008), No. 2, 67-80, where no reference is made to any gap or other issue in Mišík's work).