Related papers: Spectral dimension of $p$-adic integers

We express the classical $p$-adic integers $\hat{\mathbb{Z}_p}$, as a metric space, as a final colagebra to a certain endofunctor. We realize the addition and the multiplication on $\hat{\mathbb{Z}_p}$ as the coalgebra maps from…

The aim of this paper is to study the dimensions and standard part maps between the field of $p$-adic numbers ${{\mathbb Q}_p}$ and its elementary extension $K$ in the language of rings $L_r$. We show that for any $K$-definable set…

We give a brief overview of the theory of complex dimensions of real (archimedean) fractal strings via an illustrative example, the ordinary Cantor string, and a detailed survey of the theory of p-adic (nonarchimedean) fractal strings and…

We show that separable, nuclear and strongly purely infinite C*-algebras have finite nuclear dimension. In fact, the value is at most three. This exploits a deep structural result of Kirchberg and R{\o}rdam on strongly purely infinite…

We introduce a notion of p-dimensional cones made of $p$-dimensional subspaces and gauges on these cones, giving rise to a contraction principle which generalizes the one for Birkhoff cones. Using tools on the grassmannian and the exterior…

The problem of computing the dimension of a left/right ideal in a group algebra F[G] of a finite group G over a field F is considered. The ideal dimension is related to the rank of a matrix originating from a regular left/right…

Quantum isometry groups of spectral triples associated with approximately finite-dimensional C*-algebras are shown to arise as inductive limits of quantum symmetry groups of corresponding truncated Bratteli diagrams. This is used to…

We introduce the point degree spectrum of a represented space as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees, and so on. The notion of point degree spectrum…

Let $K/k$ be a pro-$p$-extension over a number field $k$ whose Galois group is finitely generated and $k_0\subseteq k_1\subseteq\cdots\subseteq k_n\subseteq\cdots$ an ascending sequence of intermediate fields of $K/k$ such that $k_n/k$ is…

The paper is devoted to the notion of a spectral section introduced by Melrose and Piazza. In the first part of the paper we generalize results of Melrose and Piazza to arbitrary base spaces, not necessarily compact. The second part…

Explicit formulas are obtained for a family of continuous mappings of p-adic numbers $\Qp$ and solenoids $\Tp$ into the complex plane $\sC$ and the space \~$\Rs ^{3}$, respectively. Accordingly, this family includes the mappings for which…

We extend the result of Blumberg and Mandell on K-theoretic Tate-Poitou duality at odd primes which serves as a spectral refinement of the classical arithmetic Tate-Poitou duality. The duality is formulated for the $K(1)$-localized…

Let S(n,k) denote the Stirling numbers of the second kind. We prove that the p-adic limit of S(p^e a + c, p^e b + d) as e goes to infinity exists for all integers a, b, c, and d. We call the limiting p-adic integer S(p^\infty a + c,…

A well-known result, due to Meyer, states that the set P of Pisot numbers, generating a real algebraic number field K, is uniformly discrete and relatively dense in the set of positive real number. In the present paper, we show that P is…

For the p-adic group G=SL (2) , we present results of the computations of the sums of the Bernstein projectors of a given depth. Motivation for the computations is based on a conversation with Roger Howe in August 2013. The computations are…

Let $G$ be an $r$-uniform hypergraph of order $n$. For each $p\geq 1$, the $p$-spectral radius $\lambda^{(p)}(G)$ is defined as \[ \lambda^{(p)}(G):=\max_{|x_1|^p+\cdots+|x_n|^p=1} r\sum_{\{i_1,\ldots,i_r\}\in E(G)}x_{i_1}\cdots x_{i_r}. \]…

Let p be an odd prime. Let K_p = Q(zeta) be the p-cyclotomic field. Let pi be the prime ideal of K_p lying over p. Let G be the Galois group of K_p. Let v be a primitive root mod p. Let sigma be a Q-isomorphism of K_p. Let P(sigma) =…

We introduce the nuclear dimension of a C*-algebra; this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank. Our notion behaves well with respect to inductive…

We explain the notion of minimality for an equivariant spectral triple and show that the triple for the quantum SU(2) group constructed by Chakraborty and Pal is minimal. We also give a decomposition of the spectral triple constructed by…

The goal of this paper is to give a numerical criterion for an open question in $p$-adic Fourier theory. Let $F$ be a finite extension of $\mathbf{Q}_p$. Schneider and Teitelbaum defined and studied the character variety $\mathfrak{X}$,…