A uniform constant-factor improvement on $t_p \le p-c$ for covering systems with one repeated odd prime modulus (Question 5.3 of Harrington–Sun–Wong)

For an odd prime $p$ , let $t_p$ be the smallest number of times $p$ must be allowed to appear as a modulus in a covering system of the integers in which every other modulus is odd, distinct, and greater than 1. The best known bounds are linear in $p$ with leading constant 1: currently $t_p \le p-5$ for all primes $p \ge 17$ . This problem asks whether there is a constant $0 \le \epsilon < 1$ such that $t_p \le \epsilon p$ for all sufficiently large primes $p$ — a uniform constant-factor improvement over the known bounds, posed as Question 5.3 of Harrington–Sun–Wong's "Covering systems with odd moduli" (Discrete Math. 345 (2022) 112936). The question is a quantitative stepping stone toward the Erdős odd covering problem: an odd covering system (all moduli odd, distinct, $>1$ ) would make $t_p = 0$ for all sufficiently large $p$ , so $\epsilon = 0$ would work.

Background

A covering system of the integers is a finite collection of congruence classes $r_i \pmod{m_i}$ (with $m_i \ge 2$ ) such that every integer satisfies at least one congruence in the collection; it is called distinct if all moduli $m_i$ are different. Covering systems were introduced by Erdős in 1950, and much of the subject concerns which restrictions on the moduli are compatible with covering. A landmark negative result of Hough (2015) shows that in any distinct covering system the smallest modulus is bounded by an absolute constant. Arguably the most famous still-open question in the area is the odd covering problem: does there exist a covering system whose moduli are all odd, distinct, and greater than 1? Such a system is called an odd covering; it is not known whether one exists, and known necessary conditions (for example, a 2005 result of Guo and Sun: the least common multiple of the moduli of any square-free odd covering must have at least 22 distinct prime factors) show only that such a system must be extremely large.

Problem Statement

Does there exist a constant $0 \le \epsilon < 1$ such that, for all sufficiently large primes $p$ , $t_p \le \epsilon p$ , where $t_p$ is the smallest nonnegative integer for which there exists a covering system of the integers (a finite collection of congruence classes $r_i \pmod{m_i}$ , $m_i \ge 2$ , whose union is all of the integers) having the odd prime $p$ as a modulus exactly $t_p$ times, with all other moduli odd, distinct, and greater than 1? Equivalently: can the known linear bounds $t_p \le p - c$ (best current $c = 5$ , for primes $p \ge 17$ ) be improved to $t_p \le \epsilon p$ for some uniform constant $\epsilon < 1$ and all large primes $p$ ? (This is Question 5.3 of Harrington–Sun–Wong.)

References

P. Erdős, "On integers of the form 2^k + p and some related problems", Summa Brasil. Math. 2 (1950), 113–123, MR13,437i (no DOI; scanned copy at the Alfréd Rényi Institute), https://www.renyi.hu/~p_erdos/1950-07.pdf
S. Guo and Z.-W. Sun, “On odd covering systems with distinct moduli”, Adv. in Appl. Math. 35 (2005), 182–187, DOI: 10.1016/j.aam.2005.01.004, https://doi.org/10.1016/j.aam.2005.01.004
J. Harrington, "Two questions concerning covering systems", Int. J. Number Theory 11 (2015), 1739–1750, DOI: 10.1142/S179304211550075X, https://doi.org/10.1142/S179304211550075X
B. Hough, "Solution of the minimum modulus problem for covering systems", Ann. of Math. (2) 181 (2015), 361–382, DOI: 10.4007/annals.2015.181.1.6, https://doi.org/10.4007/annals.2015.181.1.6
M. Filaseta and W. Harvey, "Covering subsets of the integers by congruences", Acta Arith. 182 (2018), 43–72, DOI: 10.4064/aa161214-4-10, https://doi.org/10.4064/aa161214-4-10
J. Harrington, Y. Sun, T. W. H. Wong, "Covering systems with odd moduli", Discrete Math. 345 (2022), Article 112936, DOI: 10.1016/j.disc.2022.112936, arXiv:2104.00602, https://arxiv.org/abs/2104.00602
R. Agrawal, P. Bhatia, K. Gupta, P. Lamb, A. Lott, A. Rice, C. R. Ward, "Computations and observations on congruence covering systems", arXiv:2208.09720 (2022), https://arxiv.org/abs/2208.09720
C. Bispels, M. Cohen, J. Harrington, J. Lowrance, K. Pontes, L. Schaumann, T. W. H. Wong, "A further investigation on covering systems with odd moduli", Discrete Math. 349 (2026), Article 115013, DOI: 10.1016/j.disc.2026.115013, arXiv:2507.16135, https://arxiv.org/abs/2507.16135