2023/08/10 by Li, Huixi, Wang, Biao, Wang, Chunlin +1
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2308.05378
In 1950, Erdős posed a question known as the minimum modulus problem on covering systems for 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 in 2015, as he proved that the minimum modulus of any covering system with distinct moduli does not exceed 1016. Recently, Balister, Bollobás, Morris, Sahasrabudhe, and Tiba 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 \mathbbFq[x] of multiplicity s is bounded by a constant depending only on s and q. Consequently, we successfully resolve the minimum modulus problem for \mathbbFq[x] and disprove a conjecture by Azlin.