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].References
- Michael Wayne Azlin, Covering Systems of Polynomial Rings Over Finite Fields, University of Mississippi, Electronic Theses and Dissertations. 39, 2011.
- P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe, and M. Tiba, Erdős covering systems, Acta Math. Hungar. 161 (2020), no. 2, 540–549. MR 4131932, DOI 10.1007/s10474-020-01048-z
- Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba, On the Erdős covering problem: the density of the uncovered set, Invent. Math. 228 (2022), no. 1, 377–414. MR 4392459, DOI 10.1007/s00222-021-01087-5
- Andreas O. Bender, Representing an element in $\mathbf {F}_q[t]$ as the sum of two irreducibles, Mathematika 60 (2014), no. 1, 166–182. MR 3164525, DOI 10.1112/S0025579313000065
- Maria Cummings, Michael Filaseta, and Ognian Trifonov, An upper bound for the minimum modulus in a covering system with squarefree moduli, arXiv:2211.08548, 2022.
- Paul Erdős, Some unsolved problems, Michigan Math. J. 4 (1957), 291–300. MR 98702
- Paul Erdős, Quelques problèmes de théorie des nombres, Monographies de L’Enseignement Mathématique, No. 6, Univ. Genève, Geneva, 1963, pp. 81–135 (French). MR 158847
- Paul Erdős, Some problems in number theory, Computers in Number Theory, Proc. Atlas Sympos. No. 2, Oxford 1969, (1971), pp. 405–414.
- Paul Erdős, Résultats et problèmes en théorie des nombres, Séminaire Delange-Pisot-Poitou (14e année: 1972/73), Théorie des nombres, Fasc. 2, Secrétariat Math., Paris, 1973, pp. Exp. No. 24, 7 (French). MR 396376
- P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], vol. 28, Université de Genève, L’Enseignement Mathématique, Geneva, 1980. MR 592420
- P. Erdös, On integers of the form $2^k+p$ and some related problems, Summa Brasil. Math. 2 (1950), 113–123. MR 44558
- Michael Filaseta, Kevin Ford, Sergei Konyagin, Carl Pomerance, and Gang Yu, Sieving by large integers and covering systems of congruences, J. Amer. Math. Soc. 20 (2007), no. 2, 495–517. MR 2276778, DOI 10.1090/S0894-0347-06-00549-2
- Carl Friedrich Gauss, Disquisitiones arithmeticae, Yale University Press, New Haven, Conn.-London, 1966. Translated into English by Arthur A. Clarke, S. J. MR 197380
- Bob Hough, Solution of the minimum modulus problem for covering systems, Ann. of Math. (2) 181 (2015), no. 1, 361–382. MR 3272928, DOI 10.4007/annals.2015.181.1.6
- Jonah Klein, Dimitris Koukoulopoulos, and Simon Lemieux, On the $j$th smallest modulus of a covering system with distinct moduli, Int. J. Number Theory 20 (2024), no. 2, 471–479. MR 4709636, DOI 10.1142/S1793042124500234
- John Knopfmacher and Wen-Bin Zhang, Number theory arising from finite fields, Monographs and Textbooks in Pure and Applied Mathematics, vol. 241, Marcel Dekker, Inc., New York, 2001. Analytic and probabilistic theory. MR 1835434, DOI 10.1201/9780203908150
- Huixi Li, Biao Wang, and Shaoyun Yi, On the minimum modulus problem in number fields, arXiv:2302.05946, 2023.
- Pace P. Nielsen, A covering system whose smallest modulus is 40, J. Number Theory 129 (2009), no. 3, 640–666. MR 2488595, DOI 10.1016/j.jnt.2008.09.016
- Michael Rosen, Number theory in function fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002. MR 1876657, DOI 10.1007/978-1-4757-6046-0
- Will Sawin and Mark Shusterman, Möbius cancellation on polynomial sequences and the quadratic Bateman-Horn conjecture over function fields, Invent. Math. 229 (2022), no. 2, 751–927. MR 4448995, DOI 10.1007/s00222-022-01115-y
- Will Sawin and Mark Shusterman, On the Chowla and twin primes conjectures over $\Bbb {F}_q[T]$, Ann. of Math. (2) 196 (2022), no. 2, 457–506. MR 4445440, DOI 10.4007/annals.2022.196.2.1
- Pari/GP version 2.5.5. The PARI Group, available from http://pari.math.u-bordeaux.fr. Bordeaux, 2013.
- Biao Wang, Dynamics on the number of prime divisors for additive arithmetic semigroups, Finite Fields Appl. 81 (2022), Paper No. 102029, 28. MR 4397757, DOI 10.1016/j.ffa.2022.102029
- R. Warlimont, Arithmetical semigroups. II. Sieving by large and small prime elements. Sets of multiples, Manuscripta Math. 71 (1991), no. 2, 197–221. MR 1101269, DOI 10.1007/BF02568402
- André Weil, Sur les courbes algébriques et les variétés qui s’en déduisent, Hermann & Cie, Paris, 1948.
Bibliographic Information
- Huixi Li
- Affiliation: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
- ORCID: 0000-0003-4447-1157
- Email: lihuixi@nankai.edu.cn
- Biao Wang
- Affiliation: School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan 650091, People’s Republic of China
- MR Author ID: 1397071
- ORCID: 0000-0003-1105-6299
- Email: bwang@ynu.edu.cn
- Chunlin Wang
- Affiliation: School of Mathematical Sciences, Sichuan Normal University, Chengdu 610064, People’s Republic of China
- MR Author ID: 1029427
- Email: c-l.wang@outlook.com
- Shaoyun Yi
- Affiliation: School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, People’s Republic of China
- MR Author ID: 1404024
- ORCID: 0000-0002-1251-8611
- Email: yishaoyun926@xmu.edu.cn
- Received by editor(s): August 29, 2023
- Received by editor(s) in revised form: November 26, 2023, and February 23, 2024
- Published electronically: July 1, 2024
- Additional Notes: The research of the first author was partially supported by the National Natural Science Foundation of China (Grant No. 12201313) and the Fundamental Research Funds for the Central Universities, Nankai University (Grant No. 63231145). The fourth author was supported by the National Natural Science Foundation of China (No. 12301016) and the Fundamental Research Funds for the Central Universities (No. 20720230025).
The second author is the corresponding author. - Communicated by: Amanda Folsom
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3731-3742
- MSC (2020): Primary 05B40, 11A07, 11B25, 11T06, 11T55
- DOI: https://doi.org/10.1090/proc/16864
- MathSciNet review: 4781969