On the minimum modulus problem in number fields

TL;DR

This paper extends the distortion method using probability measures to number fields, providing a solution to Erdős' minimum modulus problem in this broader mathematical setting.

Contribution

It generalizes the distortion method with probability measures to number fields, solving Erdős' minimum modulus problem in this context.

Findings

The minimum modulus in number fields is bounded.

The distortion method can be adapted to number fields.

A new theoretical framework for number fields is developed.

Abstract

The minimum modulus problem on covering systems was posed by Erd\H{o}s in 1950, 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 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 this method. In this paper, we develop the distortion method by introducing the theory of probability measures associated to an inverse system. Following Klein et al.'s work, we derive an analogue of their theorem in the setting of number fields, which provides a solution to Erd\H{o}s' minimum modulus problem in number fields.