New 5-adic Cantor sets and fractal string

2011

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 their complex dimensions. Moreover, we present an explicit volume formula for the tubular neighborhood of a p-adic fractal string Lp, expressed in terms of the underlying complex dimensions. Special attention will be focused on p-adic self-similar strings, in which the nonarchimedean theory takes a more natural form than its archimedean counterpart. In contrast with the archimedean setting, all p-adic self-similar strings are lattice and hence, their complex dimensions (as well as their zeros) are periodically distributed along finitely many vertical lines. The general theory is illustrated by some simple examples, the nonarchimedean Cantor, Euler, and Fibonacci strings. Throughout this comparative survey of the archimedean and nonarchimedean theor...

P-Adic Numbers, Ultrametric Analysis, and Applications, 2009

We develop a geometric theory of self-similar p-adic fractal strings and their complex dimensions. We obtain a closed-form formula for the geometric zeta functions and show that these zeta functions are rational functions in an appropriate variable. We also prove that every self-similar p-adic fractal string is lattice. Finally, we define the notion of a nonarchimedean self-similar set and discuss its relationship with that of a self-similar p-adic fractal string. We illustrate the general theory by two simple examples, the nonarchimedean Cantor and Fibonacci strings.

Journal of Number Theory, 1990

We answer the following question: If p and 4 are positive integers greater than 1 and C,, is the set of all numbers in [0, 1 ] which can be expressed in base p without using a nonempty finite collection of finite length patterns in Z,, under what conditions does C, contain a number whose base q expansion contains all patterns of finite length? 0 1990 Academic press, Inc. LEMMA 1. If there exist two nonsimilar freely allowable patterns, then C, contains a Cantor set.

2002

Beginning with the most general fractal strings/sprays construction recently expounded in the book by Lapidus and Frankenhuysen, it is shown how the complexified extension of El Naschie's Cantorian-Fractal spacetime model belongs to a very special class of families of fractal strings/sprays whose scaling ratios are given by suitable pinary (pinary, $p$ prime) powers of the Golden Mean. We then proceed

2016

The Cantor set is well-known to students of topology and analysis as both a go-to source for counterexamples and an extremely interesting, yet strange, space. This set is named after mathematician Georg Cantor (1845–1918), who constructed it in an 1883 paper “Über unendliche, lineare Punktmannigfaltigkeiten, [Part] 5” (“On infinite linear manifolds of points, Part 5”) [Cantor, 1883]. However, this set appeared as early as 1875 in a paper by the Irish mathematician Henry John Stephen Smith (1826–1883). Smith, who is best known today for introducing the Smith normal form of a matrix, was a professor at Oxford who made great contributions in matrix theory and number theory. In this project, we will explore parts of a paper he wrote entitled “On the Integration of Discontinuous Function” [Smith, 1875].

Fractals-complex Geometry Patterns and Scaling in Nature and Society, 2009

The formulation of a new analysis on a zero measure Cantor set $C (\subset I=[0,1])$ is presented. A non-archimedean absolute value is introduced in $C$ exploiting the concept of {\em relative} infinitesimals and a scale invariant ultrametric valuation of the form $\log_{\varepsilon^{-1}} (\varepsilon/x) $ for a given scale $\varepsilon>0$ and infinitesimals $0<x<\varepsilon, x\in I\backslash C$. Using this new absolute value, a valued (metric) measure is defined on $C $ and is shown to be equal to the finite Hausdorff measure of the set, if it exists. The formulation of a scale invariant real analysis is also outlined, when the singleton $\{0\}$ of the real line $R$ is replaced by a zero measure Cantor set. The Cantor function is realised as a locally constant function in this setting. The ordinary derivative $dx/dt$ in $R$ is replaced by the scale invariant logarithmic derivative $d\log x/d\log t$ on the set of valued infinitesimals. As a result, the ordinary real valued functions are expected to enjoy some novel asymptotic properties, which might have important applications in number theory and in other areas of mathematics.

2021

The ternary Cantor set C, constructed by George Cantor in 1883, is the best known example of a perfect nowhere-dense set in the real line. The present article we study the basic properties C and also study in detail the ternary expansion characterization C. We then consider the Cantor-Lebesgue function defined on C, prove its basic properties and study its continuous extension to [0, 1]. We also consider the geometric construction of F as the uniform limit of polygonal functions. Finally, we consider the Lebesgue’s function defined from C onto [0, 1]2 and onto [0, 1]3, as well as their continuous extension to [0, 1], i.e., obtained the Lebesgue’s space filling curves. Finally we discuss Hausdorff’s theorem, which is a natural generalization of the definition of Lebesgue’s functions, that states that any compact metric space is a continuous image of the Cantor set C. This notes are the outgrowth of a (zoom)-seminar during the 2020 spring and summer semesters based on H. Sagan’s book ...

Contemporary Mathematics, 2013

The theory of p-adic fractal strings and their complex dimensions was developed by the first two authors in [17, 18, 19], particularly in the self-similar case, in parallel with its archimedean (or real) counterpart developed by the first and third author in [28]. Using the fractal tube formula obtained by the authors for p-adic fractal strings in [20], we present here an exact volume formula for the tubular neighborhood of a p-adic self-similar fractal string Lp, expressed in terms of the underlying complex dimensions. The periodic structure of the complex dimensions allows one to obtain a very concrete form for the resulting fractal tube formula. Moreover, we derive and use a truncated version of this fractal tube formula in order to show that Lp is not Minkowski measurable and obtain an explicit expression for its average Minkowski content. The general theory is illustrated by two simple examples, the 3-adic Cantor string and the 2-adic Fibonacci strings, which are nonarchimedean analogs (introduced in [17, 18]) of the real Cantor and Fibonacci strings studied in [28].

Contemporary Mathematics, 1999

We put the theory of Dirichlet series and integrals in the geometric setting of`fractal strings' (one-dimensional drums with fractal boundary). The poles of a Dirichlet series thus acquire the geometric meaning of`complex dimensions' of the associated fractal string, and they describe the geometric and spectral oscillations of this string by means of an`explicit formula'. We de ne the`spectral operator', which allows us to characterize the presence of critical zeros of zeta-functions from a large class of Dirichlet series as the question of invertibility of this operator. We thus obtain a geometric reformulation of the generalized Riemann Hypothesis, thereby extending the earlier work of the rst author with H. Maier. By considering the restriction of this operator to the subclass of`generalized Cantor strings', we prove that zeta-functions from a large subclass of this class have no in nite sequence of zeros forming a vertical arithmetic progression. (For the special case of the Riemann zeta-function, this is Putnam's theorem.) We make an extensive study of the complex dimensions of`self-similar' fractal strings, to gain further insight into the kind of geometric information contained in the complex dimensions. We also obtain a formula for the volume of the tubular neighborhoods of a fractal string and draw an analogy with Riemannian geometry. Our work suggests to de ne`fractality' as the presence of nonreal complex dimensions with positive real part.

This paper is intended to give exposition on The Cantor Set: its properties, Construction and its measurability. The key objective is to point to the beginner and others interested in the study of this famous and fascinating set, The Cantor Set, of some of the main ideas with regards to perfect, Compact but not empty sets which further should be researched into. This paper serves as an accessible introduction to beginners and other scholars interested in the studying the Cantor set especially its properties and measurability.