Spectral dimension of $p$-adic integers

TL;DR

This paper establishes that the spectral dimension of the ring of p-adic integers is zero, matching its manifold dimension, and explores its K-theory, revealing generators as characters.

Contribution

It proves the spectral dimension of p-adic integers is zero and characterizes the K-groups, including generators expressed as characters.

Findings

Spectral dimension of $p$-adic integers is zero.

K-groups of $p$-adic integers are determined.

Generators of $K_0(p)$ are finite spans of characters.

Abstract

The notion of spectral dimension was introduced by Chakraborty and Pal in \cite{cp}. In this paper, we show that the spectral dimension of the ring of $p$-adic integers, $\mathbb{Z}_p$, is equal to its manifold dimension, which is $0$. Finally, we determine the $K$-groups of $\mathbb{Z}_p$, and show that the generators of $K_0(\mathbb{Z}_p)$ can be expressed as finite span of the characters of $\mathbb{Z}_p$.