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Claim. When $X_n=\{0\}$ for every $n\in A$, the set $B$ of Problem 486 has a logarithmic density, for every $A\subseteq\mathbb N$. In this case $B$ is the set of integers that are not proper multiples of a member of $A$. Theorem 1(a) of H. Davenport and P. Erdős, On sequences of positive integers, Acta Arith. 2 (1936), 147–151 (card), proves that the set of all multiples of a sequence $a_1,a_2,\ldots$ has logarithmic density $\mathcal A$, the limit of the inclusion-exclusion densities of the multiples of its first $m$ terms; the proof writes the Dirichlet series of the set's indicator as $\zeta(s)$ times a series whose finite approximants are monotone in $s$ and applies a Tauberian theorem of Hardy and Littlewood. The set of all multiples of $A$ differs from the set of proper multiples by the members of $A$ divisible by no smaller member, a primitive set, and by Behrend's theorem the reciprocal sum of a primitive set up to $x$ is $O(\log x/\sqrt{\log\log x})$, so that difference has logarithmic density zero (the bound is stated on the card of Erdős, Sárközy and Szemerédi's sharpening). Hence $B$ has logarithmic density $1-\mathcal A$. The same authors' second paper of the same title, J. Indian Math. Soc. (N.S.) 15 (1951), 19–24 (card), linked above, replaces the Tauberian argument by a direct elementary proof through the integers supported on the first $k$ primes.

Covers. Every instance in which every $X_n$ is the zero class, for every choice of $A$; for these the answer is yes. Not covered: any instance with a nonzero residue in some $X_n$, including the singleton case $|X_n|=1$ of Problem 25, which the site's remark says this problem generalizes, and the general case, for which Wang's 2026 manuscript on its own claim page claims a disproof. Besicovitch's examples, cited in the site's remark, show that natural density can fail even when $X_n=\{0\}$ for all $n$, which is why the problem asks for logarithmic density.

Depends on. No page of this wiki.

Acceptance. The refereed evidence is the 1936 journal publication in Acta Arithmetica, with the 1951 elementary proof in the Journal of the Indian Mathematical Society. The site labels the problem OPEN and its remark (page last edited 8 April 2026) credits both papers with the case $X_n=\{0\}$; a remark on an open problem is not an acceptance, so no reviewed evidence is listed. The records give the publication years and no finer dates, so the page carries the first day of 1936.