Spectral dimension of $p$-adic integers

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 ${\mathbb{Z}}_p$.