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 $\mathbb{ Z}_p$.
Categories: math.QA