Problem 281

Let $n_1<n_2<\cdots$ be an infinite sequence such that, for any choice of congruence classes $a_i\pmod{n_i}$ , the set of integers not satisfying any of the congruences $a_i\pmod{n_i}$ has density $0$ .

Is it true that for every $\epsilon>0$ there exists some $k$ such that, for every choice of congruence classes $a_i$ , the density of integers not satisfying any of the congruences $a_i\pmod{n_i}$ for $1\leq i\leq k$ is less than $\epsilon$ ?

Source: erdosproblems.com/281

Number theory
Covering systems

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An accepted solution exists. The statement is true.

PROVED (LEAN). Two independent arguments posted on the site's thread in January 2026 answer the question yes and are credited in the site's commentary: Neel Somani's proof, produced with GPT-5.2 Pro, through Haar measure on the profinite integers and Dini's theorem (claim page (Somani, 2026)), and KoishiChan's elementary proof from the Davenport-Erdős theorem on sets of multiples and Rogers' theorem on zero residues (claim page (KoishiChan, 2026)); both are accepted on the curator's credit, and neither is refereed. The site's Lean qualification refers to the Lean formalization of Somani's argument recorded under Formalization.

Problem 281

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