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

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 $\{0,1,2,\cdots ,p-1{\}}^{\mathbb{Z}}$.

24 pages