Explicit effective bounds for covering systems of the polynomial ring $\mathbb{F}_q[x]$: making the minimum-degree constants of Li–Wang–Wang–Yi quantitative

Li, Wang, Wang and Yi (Proc. AMS 2024) proved the function-field analogue of Erdős's minimum modulus problem: in any covering system of $\mathbb{F}_q[x]$ of multiplicity $s$, the smallest degree of a modulus is at most $3(c+3\log_q s)\,\log(cs+3s\log_q s)$ for an absolute constant $c>0$ that is not explicitly specified, and the degree of the $n$-th smallest modulus in a minimal covering system with distinct moduli is bounded, but only qualitatively. The open problem is to make these results effective: determine an explicit value for $c$ (any explicitly computable $C(q,s)$ suffices), obtain an explicit, ideally small numerical bound for the distinct case in the non-vacuous range $q\le 73$ (where such systems exist), and derive an explicit bound $B_n(q)$ for the $n$-th smallest modulus degree. Follow-up work has established non-existence of covering systems for large $q$ and a Crittenden–Vanden Eynden covering criterion in $\mathbb{F}_q[x]$, but no explicit constants or quantitative $n$-th-smallest-degree bounds are known.

Background

A covering system of the integers is a finite collection of arithmetic progressions (congruence classes) whose union is all of $\mathbb{Z}$. In 1950 Erdős asked the minimum modulus problem: is the smallest modulus of a covering system whose moduli are pairwise distinct necessarily bounded? Hough (Ann. of Math. 2015) resolved this, proving the minimum modulus is at most $10^{16}$; Balister, Bollobás, Morris, Sahasrabudhe and Tiba later developed a versatile probabilistic–sieve technique, the distortion method, and reduced the bound to $616{,}000$. Related integer results bound the $j$-th smallest modulus (Klein, Koukoulopoulos, Lemieux) and the squarefree-moduli case (Cummings, Filaseta, Trifonov: minimum modulus $\le 118$).

The natural function-field analogue replaces $\mathbb{Z}$ by the polynomial ring $\mathbb{F}_q[x]$ over the finite field with $q$ elements. Here an arithmetic progression is a coset $a(x)+\langle d(x)\rangle$, where $\langle d(x)\rangle=d(x)\mathbb{F}_q[x]$ is the ideal of multiples of the modulus $d(x)$, which as in Li–Wang–Wang–Yi (Proc. AMS 2024) is nonconstant, i.e. of positive degree ($|d(x)|=q^{\deg d(x)}>1$, so that the degenerate constant moduli are excluded); a finite collection of such cosets covering $\mathbb{F}_q[x]$ is a covering system, its multiplicity is the maximum number of times any single modulus is repeated, and it has distinct moduli when the multiplicity is $1$. A covering system is minimal if no proper subcollection still covers. Azlin conjectured (PhD thesis, 2011) that for every $D$ there is a distinct covering system of $\mathbb{F}_q[x]$ in which every modulus has degree at least $D$.

Li, Wang, Wang and Yi (Proc. AMS 2024) disproved Azlin's conjecture by adapting the distortion method to $\mathbb{F}_q[x]$. Their Theorem 1.4 states that for every covering system of $\mathbb{F}_q[x]$ of multiplicity $s$ with moduli $d_1(x),\dots,d_k(x)$,

$$\min_{1\le i\le k}\deg d_i(x)\;\le\;3(c+3\log_q s)\,\log(cs+3s\log_q s),$$

for an absolute constant $c>0$, with $\log$ denoting the natural logarithm, which, as the authors emphasize, is not explicitly specified — the argument (following Klein–Koukoulopoulos–Lemieux) does not track constants. They note that one could instead try to carry out the explicit distortion-method computations of Balister et al. in the $\mathbb{F}_q[x]$ setting to get a small numerical bound for distinct covering systems, as in the integer case. Their Theorem 1.5, deduced from Theorem 1.4 by an induction on $n$, states only qualitatively that in a minimal covering system of $\mathbb{F}_q[x]$ with distinct moduli the degree of the modulus with the $n$-th smallest degree is bounded by a constant depending on $n$ and $q$; they propose that a suitable generalisation to $\mathbb{F}_q[x]$ of the Crittenden–Vanden Eynden theorem (any $n$ arithmetic progressions of $\mathbb{Z}$ covering the first $2^n$ positive integers cover all integers) could yield a quantitative form, but no such generalisation was then available.

Subsequent work has reshaped the landscape without making anything effective. Li–Wang–Wang–Yi (JNT 2025) proved that no covering system of multiplicity $s$ exists in any global function field (finite extension of $\mathbb{F}_q(x)$) of genus $g$ once $q\ge(1.14+0.16g)e^{6.5+0.97g}s^2$, in particular no distinct covering system of $\mathbb{F}_q[x]$ exists for $q\ge 759$; Biao Wang (IJNT 2026) improved this to $q\ge(82.26+18.88g)e^{0.95g}s^2$ and $q>73$ for the distinct case over $\mathbb{F}_q[x]$. Separately, Rongyin Wang proved the Crittenden–Vanden Eynden analogue in $\mathbb{F}_q[x]$: any $n$ congruence classes that fail to cover $\mathbb{F}_q[x]$ omit some polynomial of degree less than $n$ (with the sharp scale conjectured and then asymptotically established as $n/(q-1)+O_q(1)$ in a 2026 follow-up). None of these works determines the constant $c$, produces any explicit numerical minimum-degree bound, or derives a quantitative version of Theorem 1.5.

Problem Statement

Make the ineffective and qualitative boundedness theorems of Li–Wang–Wang–Yi for covering systems of $\mathbb{F}_q[x]$ effective and quantitative. The single common objective is to replace constants that are merely known to exist by explicitly computable bounds; it comprises three linked deliverables:

1. Effective Theorem 1.4. Determine an explicit numerical value for the absolute constant $c>0$ in the bound

$$\min_{1\le i\le k}\deg d_i(x)\le 3(c+3\log_q s)\,\log(cs+3s\log_q s)$$

of Li–Wang–Wang–Yi (for a covering system of multiplicity $s$ with moduli $d_1(x),\dots,d_k(x)$); it suffices for this deliverable to prove some explicitly computable function $C(q,s)$ such that every covering system of $\mathbb{F}_q[x]$ of multiplicity $s$ contains a modulus $d_i(x)$ with $\deg d_i(x)\le C(q,s)$.

2. Explicit distinct case. For covering systems of $\mathbb{F}_q[x]$ with distinct moduli ($s=1$), which by Biao Wang's non-existence theorem can exist only for $q\le 73$, obtain an explicit numerical bound (as a function of $q$) on the smallest modulus degree, ideally small, of the kind Hough's $10^{16}$ and the bound $616{,}000$ of Balister–Bollobás–Morris–Sahasrabudhe–Tiba provide for $\mathbb{Z}$.

3. Quantitative Theorem 1.5. Prove an explicitly computable bound $B_n(q)$ for the degree of the modulus with the $n$-th smallest degree in any minimal covering system of $\mathbb{F}_q[x]$ with distinct moduli that has at least $n$ moduli (the moduli ordered by non-decreasing degree, as in Section 5 of the source paper), as an explicit function of $n$ and $q$.

The deliverables are scientifically linked: the induction in Section 5 of the source paper converts any explicit constant from deliverable 1 into an explicit (recursively defined) bound in deliverable 3. Each bound obtained must be explicit and cover the full hypotheses stated (all prime powers $q$ for deliverable 1; all $q\le 73$ for deliverables 2 and 3).

References

1. Huixi Li, Biao Wang, Chunlin Wang, Shaoyun Yi, On covering systems of polynomial rings over finite fields, Proceedings of the American Mathematical Society 152 (2024), no. 9, 3731–3742. DOI: 10.1090/proc/16864. https://doi.org/10.1090/proc/16864 (arXiv:2308.05378, https://arxiv.org/abs/2308.05378)
2. Huixi Li, Biao Wang, Chunlin Wang, Shaoyun Yi, On Erdős covering systems in global function fields, Journal of Number Theory 266 (2025), 269–280. DOI: 10.1016/j.jnt.2024.07.002. https://doi.org/10.1016/j.jnt.2024.07.002 (arXiv:2402.03810, https://arxiv.org/abs/2402.03810)
3. Biao Wang, Bounds for Erdős covering systems in global function fields, International Journal of Number Theory 22 (2026), no. 6, 1327–1336. DOI: 10.1142/S1793042126500703. https://doi.org/10.1142/S1793042126500703 (arXiv:2408.10460, https://arxiv.org/abs/2408.10460)
4. Rongyin Wang, On an Erdős-type conjecture on $\mathbb{F}_q[x]$, preprint (2024, revised 2025). arXiv:2407.15146. https://arxiv.org/abs/2407.15146
5. Rongyin Wang, An Asymptotic Bound for Non-covering Congruence Systems over $\mathbb{F}_q[x]$, preprint (2026). arXiv:2607.27538. https://arxiv.org/abs/2607.27538
6. Bob Hough, Solution of the minimum modulus problem for covering systems, Annals of Mathematics (2) 181 (2015), no. 1, 361–382. DOI: 10.4007/annals.2015.181.1.6. https://doi.org/10.4007/annals.2015.181.1.6
7. Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, Marius Tiba, On the Erdős covering problem: the density of the uncovered set, Inventiones mathematicae 228 (2022), no. 1, 377–414. DOI: 10.1007/s00222-021-01087-5. https://doi.org/10.1007/s00222-021-01087-5
8. Jonah Klein, Dimitris Koukoulopoulos, Simon Lemieux, On the jth smallest modulus of a covering system with distinct moduli, International Journal of Number Theory 20 (2024), no. 2, 471–479. DOI: 10.1142/S1793042124500234. https://doi.org/10.1142/S1793042124500234
9. R. B. Crittenden, C. L. Vanden Eynden, Any n arithmetic progressions covering the first $2^n$ integers cover all integers, Proceedings of the American Mathematical Society 24 (1970), no. 3, 475–481. DOI: 10.1090/s0002-9939-1970-0258719-2. https://doi.org/10.1090/s0002-9939-1970-0258719-2
10. Maria Cummings, Michael Filaseta, Ognian Trifonov, An upper bound for the minimum modulus in a covering system with squarefree moduli, preprint (2022). arXiv:2211.08548. https://arxiv.org/abs/2211.08548
11. Michael Wayne Azlin, Covering Systems of Polynomial Rings Over Finite Fields, PhD thesis, University of Mississippi, 2011. https://egrove.olemiss.edu/etd/39