Problem 486

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claims/: The 2 claim pages of Problem 486, one per claimant's result; the problem's standing derives from them.

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Statement. Let $A\subseteq \mathbb{N}$, and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$. Let

$$B = \{ m\in \mathbb{N} : m\not\in X_n\pmod{n}\textrm{ for all }n\in A\textrm{ with }m>n\}.$$

Must $B$ have a logarithmic density, i.e. is it true that

$$\lim_{x\to \infty} \frac{1}{\log x}\sum_{\substack{m\in B\\ m<x}}\frac{1}{m}$$

exists?

Status. OPEN, the site's label (2026-10-06). The derived standing departs from it because a 2026 manuscript of Wang, produced with GPT-5.6 Sol, claims a disproof by a fixed delayed congruence system whose survivor set has no logarithmic density; it is a pending full claim on its claim page, which makes the problem claimed as disproved, and it has no refereed publication, site acceptance or review by a named reviewer. The case $X_n=\{0\}$, which the site's remark credits to Davenport and Erdős, is an accepted partial claim on its claim page.

Source. erdosproblems.com/486, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #486, https://www.erdosproblems.com/486.

References.

• [Be34] Besicovitch, A., On the density of certain sequences of integers. Math. Annalen (1934), 336-341.
• [DaEr36] Davenport, H. and Erdős, P., On sequences of positive integers. Acta Arithmetica (1936), 147-151.
• [DaEr51] Davenport, H. and Erdős, P., On sequences of positive integers. J. Indian Math. Soc. (N.S.) (1951), 19-24.
• [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

Formalization. Statement in formal-conjectures. Since 18 September 2026 that file, as of the commit of that date, has tagged the problem research solved, crediting Wang, and linked two formal proofs: Boris Alexeev's Lean 4.33.0 port of Wang's own formalization, and a fork of formal-conjectures that carries Wang's development through Alexeev's port and adds a bridge to the catalog's statement. Wang's development, Alexeev's port and the fork are linked at pinned commits from the claim page. This corpus has built and audited none of them, so none supplies formalized evidence; the catalog links a formal proof and does not referee it.

Current assessment

The site labels the problem OPEN (2026-10-06; commentary last edited 8 April 2026), and the standing follows the claim pages. The outstanding claim is Wang's 2026 manuscript, titled a proposed solution, whose Theorem 1.1 constructs fixed infinite $A$ and fixed $X_n$ for which the logarithmic averages of the survivor set $B$ have liminf at most $177/200$ and limsup at least $49/50$, so that $B$ has no logarithmic density; if correct, this disproves the exact question above. The claim was posted to the site's proof-claims tab on 16 July 2026 with a Lean development following on 17 July; the curator wrote that he had not checked the proof and would wait for a formalized version, a forum commenter reported a kernel replay of the development on 2 August 2026, and formal-conjectures has linked two formal proofs since 18 September 2026. None of these is a refereed publication, a site acceptance or a review by a named reviewer, and no build or audit of any of the Lean developments by this corpus is recorded, so the claim stays claimed and the problem's standing is claimed, disproved, as the claim page records.

The positive case $X_n=\{0\}$ for all $n\in A$, in which $B$ is the set of integers that are not proper multiples of a member of $A$, is settled by Davenport and Erdős's 1936 theorem that a set of multiples has a logarithmic density, together with Behrend's theorem for the primitive set by which the proper multiples differ from all multiples; it is recorded as an accepted partial claim with refereed evidence on its claim page, with the authors' elementary proof of 1951 as a second publication. Wang's construction uses many residues per modulus and lies outside this case.

Progress

Wang's proposed counterexample uses finite probabilistic deletion blocks. At each sufficiently large dyadic scale, many integers in a short interval are deleted by already-active moduli, while the completed periodic footprint of those residue classes is stretched-exponentially small. A gliding-hump construction places long runs of these blocks between recovery gaps and forces two distinct limiting regimes for the logarithmic averages.

Known Results

Davenport and Erdős (1936, Theorem 1(a); 1951, elementary proof): the set of multiples of any sequence has a logarithmic density, so $B$ has one whenever $X_n=\{0\}$ for every $n\in A$; see the claim page.

Claimed by Wang (2026, Theorem 1.1; unrefereed manuscript): there are fixed infinite $A\subseteq\mathbb N$ and fixed $X_n\subseteq\mathbb Z/n\mathbb Z$ such that the associated survivor set $B$ satisfies

$$\liminf_{x\to\infty}\frac1{\log x} \sum_{\substack{m<x\\m\in B}}\frac1m \leq\frac{177}{200} <\frac{49}{50} \leq \limsup_{x\to\infty}\frac1{\log x} \sum_{\substack{m<x\\m\in B}}\frac1m.$$

In particular, $B$ would have no logarithmic density.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• davenport_1951_sequences_positive_integers
• davenport_1951_sequences_positive_integers / main_theorem
• davenport_1936_sequences_positive_integers
• wang_2026_proposed_solution_erdos_problem_486