A density Chinese Remainder Theorem

TL;DR

This paper explores conditions under which pairs of residue classes modulo different integers have solutions to simultaneous congruences, with applications to the Lonely Runner Conjecture.

Contribution

It introduces new criteria for residue class collections to guarantee solutions to linear systems, extending classical Chinese Remainder Theorem results.

Findings

Established conditions for the existence of solutions for residue class pairs.

Quantified the number of admissible pairs (a,b) satisfying the system.

Applied results to cases where A and B are intervals, relevant to the Lonely Runner Conjecture.

Abstract

Given collections A and B of residue classes modulo m and n, respectively, we investigate conditions on A and B that ensure that, for at least some a in A and b in B, the linear system x = a mod m, x = b mod n has an integer solution, and we quantify the number of such admissible pairs (a,b). The special case where A and B consist of intervals of residue classes has application to the Lonely Runner Conjecture.