An ergodic approach towards an equidistribution result of Ferrero--Washington

TL;DR

This paper offers a new proof of an equidistribution result crucial for understanding cyclotomic $p$-adic $L$-functions, using ergodic theory instead of the traditional Weyl criterion.

Contribution

It introduces an ergodic dynamical approach to prove an equidistribution result related to cyclotomic $p$-adic $L$-functions, providing an alternative to the Weyl criterion-based proof.

Findings

Established an ergodic skew-product map on $ Z_p \times [0,1]$

Identified the map as a factor of the Bernoulli shift on a sample space

Provided an alternative proof of the equidistribution result

Abstract

An important ingredient in the Ferrero--Washington proof of the vanishing of cyclotomic $\mu$-invariant for Kubota--Leopoldt $p$-adic $L$-functions is an equidistribution result which they established using the Weyl criterion. The purpose of our manuscript is to provide an alternative proof by adopting a dynamical approach. A key ingredient to our methods is studying an ergodic skew-product map on $\mathbb{Z}_p \times [0,1]$, which is then suitably identified as a factor of the $2$-sided Bernoulli shift on the sample space $\{0,1,2,\cdots,p-1\}^{\mathbb{Z}}$.