Erdős Problem #281 — Suppose are such that for every choice of the set of integers not satisfying any of the congruences has density 0.
Suppose are such that for every choice of the set of integers not satisfying any of the congruences has density 0. In this case we must have and, if the are pairwise relatively prime, then this suffices. This property clearly holds if for every there is a so that the density of integers not satisfying , , is less than . Is this in fact necessary?
References
Additional references
Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28 (1980).
Progress summary
The question has been settled affirmatively: classical results show that sufficiently many initial congruence classes leave arbitrarily little uncovered.
The problem asks whether pointwise density-zero coverage by an infinite sequence of moduli can be made uniform over all choices after a finite initial segment. The affirmative conclusion is attributed to Somani, with an argument also presented through classical theorems of Davenport–Erdős and Rogers.
Known results
- Davenport–Erdős: the relevant lower density equals the limit of the finite-prefix densities.
- Rogers: uncovered density is maximized when every selected residue class is zero.
Affirmative resolution
Somani’s proof discussion derives the required uniform finite-prefix conclusion from those two theorems. A Lean formalization records the argument and labels the result “research solved,” providing corroborating formal evidence.
Current status (as of March 2026): The stated problem is resolved affirmatively; only questions about possible intended variants remain separate.
Solutions 0
No solutions have been posted yet.