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arXiv · 2302.05946

On the minimum modulus problem in number fields

Huixi Li·Biao Wang·Shaoyun Yi
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Abstract

The minimum modulus problem on covering systems was posed in 1950 by Erd\H{o}s, who asked whether the minimum modulus of a covering system with distinct moduli is bounded. In 2007, Filaseta, Ford, Konyagin, Pomerance and Yu affirmed it if the reciprocal sum of the moduli of a covering system is bounded. Later in 2015, Hough resolved this problem by showing that the minimum modulus in any covering system with distinct moduli is at most $10^{16}$. In 2022, Balister, Bollob\'as, Morris, Sahasrabudhe and Tiba reduced this bound to $616,000$ by developing a versatile method called the distortion method. Recently, Klein, Koukoulopoulos and Lemieux generalized Hough's result by using a suitable modification of the distortion method. In this paper, we develop the distortion method further by introducing the theory of probability measures associated to an inverse system. Following Klein et al.'s work, we provide a solution to Erd\H{o}s' minimum modulus problem in number fields. As an application, we prove that the $j$-th smallest norm in a minimal covering system of a number field with distinct moduli is bounded.

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Huixi Li, Biao Wang, Shaoyun Yi. 2023-02-12. On the minimum modulus problem in number fields. https://arxiv.org/abs/2302.05946

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