Currently displaying 1 – 10 of 10

Showing per page

Order by Relevance | Title | Year of publication

The complex sum of digits function and primes

Jörg M. Thuswaldner — 2000

Journal de théorie des nombres de Bordeaux

Canonical number systems in the ring of gaussian integers ℤ [ i ] are the natural generalization of ordinary q -adic number systems to ℤ [ i ] . It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number b . In this paper we investigate the sum of digits function ν b of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the f -th power of a prime. Furthermore, we establish an Erdös-Kac type theorem...

Unimodular Pisot substitutions and their associated tiles

Jörg M. Thuswaldner — 2006

Journal de Théorie des Nombres de Bordeaux

Let σ be a unimodular Pisot substitution over a d letter alphabet and let X 1 , ... , X d be the associated Rauzy fractals. In the present paper we want to investigate the boundaries ∂ X i ( 1 ≤ i ≤ d ) of these fractals. To this matter we define a certain graph, the so-called contact graph 𝒞 of σ . If σ satisfies a combinatorial condition called the super coincidence condition the contact graph can be used to set up a self-affine graph directed system whose...

Topological properties of two-dimensional number systems

Shigeki Akiyama;  Jörg M. Thuswaldner — 2000

Journal de théorie des nombres de Bordeaux

In the two dimensional real vector space ℝ 2 one can define analogs of the well-known q -adic number systems. In these number systems a matrix M plays the role of the base number q . In the present paper we study the so-called fundamental domain ℱ of such number systems. This is the set of all elements of ℝ 2 having zero integer part in their “ M -adic” representation. It was proved by Kátai and Környei, that ℱ is a compact set and certain translates of it form a tiling of the ℝ 2 . We construct points, where...

On the Fundamental Group of self-affine plane Tiles

Jun Luo;  Jörg M. Thuswaldner — 2006

Annales de l’institut Fourier

Let A ∈ ℤ 2 × ℤ 2 be an expanding matrix, 𝒟 ⊂ ℤ 2 a set with | det ( A ) | elements and define 𝒯 via the set equation A 𝒯 = 𝒯 + 𝒟 . If the two-dimensional Lebesgue measure of 𝒯 is positive we call 𝒯 a self-affine plane tile. In the present paper we are concerned with topological properties of 𝒯 . We show that the fundamental group π 1 ( 𝒯 ) of 𝒯 is either trivial or uncountable and provide criteria for the triviality as well as the uncountability of π 1 ( 𝒯 ) . Furthermore, we give a short proof of the fact that the closure of each component of int ( 𝒯 ) is a locally...

Fundamental groups of one-dimensional spaces

Gerhard Dorfer;  Jörg M. Thuswaldner;  Reinhard Winkler — 2013

Fundamenta Mathematicae

Let X be a metrizable one-dimensional continuum. We describe the fundamental group of X as a subgroup of its Čech homotopy group. In particular, the elements of the Čech homotopy group are represented by sequences of words. Among these sequences the elements of the fundamental group are characterized by a simple stabilization condition. This description of the fundamental group is used to give a new algebro-combinatorial proof of a result due to Eda on continuity properties of homomorphisms from...

Page 1

Download Results (CSV)