Nonarchimedean Cantor set and string

Abstract

Abstract. We construct a nonarchimedean (or p-adic) analogue of the classical ternary Cantor set C. In particular, we show that this nonarchimedean Cantor set C3 is self-similar. Furthermore, we characterize C3 as the subset of 3-adic integers whose elements contain only 0's and 2's in their 3-adic expansions and prove that C3 is naturally homeomorphic to C. Finally, from the point of view of the theory of fractal strings and their complex fractal dimensions [7,8], the corresponding nonarchimedean Cantor string resembles the standard archimedean (or real ) Cantor string perfectly.

New 5-adic Cantor sets and fractal string

“…Information about the geometry of Cantor string like Minkowski dimension and the Minkowski measurability is obtained from its geometric zeta function. Motivated by the research of Lapidus with other researcher’s (Lapidus and Hung 2008) on the Cantor string, we introduce a new Cantor one-fifth set as an example of fractal string.…”

“…The poles of the such function are the set of complex numbers (see (Lapidus and Hung 2008), pp. 7) and given by…”

“…In 2008, (Lapidus and Hung 2008; 2009) provided a framework for unifying the archimedean and p-adic (nonarchimedean) fractal string with their geometric zeta functions and complex dimensions for 3-adic Cantor sets and also the general case for p-adic Cantor sets respectively. Recently, (Ashish, Mamta Rani and Renu Chugh, Variants of Cantor Sets Using IFS, submitted and Ashish, Mamta Rani and Renu Chugh, Study of Variants of Cantor sets., submitted) studied the variants of Cantor sets and established their mathematical analysis using mathematical feedback system and iterated function system respectively.…”

Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings

“…In this paper, we further develop the geometric theory of p-adic (or nonarchimedean) fractal strings, which are bounded open subsets of the p-adic Q p with a fractal "boundary", along with the associated theory of complex dimensions and, especially, of fractal tube formulas in the nonarchimedean setting. This theory, which was first developed by Michel Lapidus and Lũ' Hùng in [37][38][39], as well as later, by those same authors and Machiel van Frankenhuijsen in [40], extends the theory of real (or archimedean) fractal strings and their complex dimensions in a natural way. Following [37][38][39][40], we introduce suitable geometric zeta functions for p-adic fractal strings whose poles play the role of complex dimensions for the standard real fractal strings.…”

Towards quantized number theory: spectral operators and an asymmetric criterion for the Riemann hypothesis

“…We note that recent developments in the theory are described in [7, ch. 13], including a first attempt at a higher dimensional theory of complex dimensions for the special case of fractal sprays (in the sense of [30]) and self-similar tilings (see [7, §13.1], based on [63][64][65][66]72]), p-adic fractal strings and associated non-Archimedean fractal tube formulae (see [7, §13.2], based on [56][57][58][59][60]), multi-fractal zeta functions and their 'tapestries' of complex dimensions (see [7, §13.3], based on [50,55,67]), random fractal strings (such as stochastically self-similar strings and the zero-set of Brownian motion) and their spectra (see [7, §13.4], based on [51]), as well as a new approach to the RH based on a conjectural Riemann flow of fractal membranes (i.e. quantized fractal strings) and correspondingly flows of zeta functions (or 'partition functions') and of the associated zeros (see [7, §13.5], which gives a brief overview of the aforementioned book [20], In search of the Riemann zeros).…”