On covering systems of polynomial rings over finite fields

Li, Huixi; Wang, Biao; Wang, Chunlin; Yi, Shaoyun

doi:10.1090/proc/16864

On covering systems of polynomial rings over finite fields
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by Huixi Li, Biao Wang, Chunlin Wang and Shaoyun Yi;

Proc. Amer. Math. Soc. 152 (2024), 3731-3742

DOI: https://doi.org/10.1090/proc/16864

Published electronically: July 1, 2024

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Abstract:

In 1950, Erdős posed a question known as the minimum modulus problem on covering systems for $\mathbb {Z}$, which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was finally resolved by Hough [Ann. of Math. (2) 181 (2015), no. 1, pp. 361–382] in 2015, as he proved that the minimum modulus of any covering system with distinct moduli does not exceed $10^{16}$. Recently, Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [Invent. Math. 228 (2022), pp. 377–414] developed a versatile method called the distortion method and significantly reduced Hough’s bound to $616,000$. In this paper, we apply this method to present a proof that the smallest degree of the moduli in any covering system for $\mathbb {F}_q[x]$ of multiplicity $s$ is bounded by a constant depending only on $s$ and $q$. Consequently, we successfully resolve the minimum modulus problem for $\mathbb {F}_q[x]$ and disprove a conjecture by Azlin [Covering Systems of Polynomial Rings Over Finite Fields, University of Mississippi, Electronic Theses and Dissertations. 39, 2011].

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On covering systems of polynomial rings over finite fields
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Abstract: