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

TL;DR

The authors address the FW equidistribution criterion, a key ingredient in proving the vanishing of the cyclotomic μ-invariant for Kubota–Leopoldt p-adic L-functions, by recasting it in ergodic terms. They construct an ergodic skew-product on the $p$-adic solenoid and relate it to the two-sided Bernoulli shift, showing that almost every point in $Z_p$ yields a jointly equidistributed digit-sum sequence in $[0,1]^r$ via a dynamical orbit. Central to the method are two propositions that reduce the problem to the ergodic behavior on the $p$-adic side and to the $r=1$ case, together with a compatibility of measures across several models. By transferring dynamics from the real-coordinate setting to the $p$-adic framework and proving a reduction to the single-variable case, they provide a dynamical proof of the equidistribution result, with potential generalizations to wider arithmetic contexts such as quotients of $ ext{SL}_2$ by $p$-adic groups. The work highlights a fruitful link between symbolic dynamics and Iwasawa-theoretic equidistribution, offering a framework that may yield broader non-commutative or automorphic applications.

Abstract

An important ingredient in the Ferrero--Washington proof of the vanishing of cyclotomic $μ$-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 $\{0,1,2,\cdots,p-1\}^{\mathbb{Z}}$.

Figures (3)

• Figure 1: The $p$-adic solenoid can be visualized as the inverse limit $\cdots \xrightarrow {p} \frac{\mathbb{R}}{\mathbb{Z}} \xrightarrow{p} \frac{\mathbb{R}}{\mathbb{Z}}\xrightarrow{p} \frac{\mathbb{R}}{\mathbb{Z}}$. In this figure, we consider $p=3$. The color gray, dark slate gray and white represent the digits $0$, $1$ and $2$ respectively. Here, the unit circle is used to represent $\mathbb{R}/\mathbb{Z}$. Here, the node represents the point $\exp\left(2\pi\times \dfrac{({\color{spanishgray}0}+{\color{darkslategray}1}\times 3)}{3^2}\right)$ on the unit circle.
• Figure 2: Cylinder sets and open intervals
• Figure 3: Visualizing the cylinder sets

Theorems & Definitions (23)

• Theorem 1: Ferrero--Washington
• Proposition 1: Iwasawa, Ferrero--Washington
• Proposition 2: Ferrero--Washington
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Proposition 1.5
• Proposition 1.6
• Remark 1.7