Jörg M. Thuswaldner

Canonical number systems, counting automata and fractals

In this paper we study properties of the fundamental domain [Fscr ] β of number systems, which are defined in rings of integers of number fields. First we construct addition automata for these number systems. Since [Fscr ] β defines a tiling of the n -dimensional vector space, we ask, which tiles of this tiling ‘touch’ [Fscr ] β . It turns out that the set of these tiles can be described with help of an automaton, which can be constructed via an easy algorithm which starts with the above-mentioned addition automaton. The addition automaton is also useful in order to determine the box counting dimension of the boundary of [Fscr ] β . Since this boundary is a so-called graph-directed self-affine set, it is not possible to apply the general theory for the calculation of the box counting dimension of self similar sets. Thus we have to use direct methods.

On the characterization of canonical number systems

It is well known that each positive integer n can be expressed uniquely as a sum n = d0 + d1b + . . . + dhb h with an integral base number b ≥ 2, dh 6= 0 and di ∈ {0, . . . , b − 1}. This concept can be generalized in several directions. On the one hand the base sequence 1, b, b, . . . can be replaced by a sequence 1 = u0 < u1 < u2 < . . . to obtain representations of positive integers. Of special interest is the case where the sequence {ui}i=0 is defined by a linear recurrence. A famous example belonging to this class is the so-called Zeckendorf representation. On the other hand, one can generalize the set of numbers which can be represented. We mention two kinds of number systems belonging to this class: The so called β-expansions introduced by Rényi [27] which are representations of real numbers in the unit interval as sums of powers of a real base number β. These digit representations of real numbers are strongly related to digit representations of positive integers if β is a zero of the characteristic polynomial of a linear recurring base sequence {ui}i=0. Of special interest is the case where β is a Pisot number. These expansions have been extensively studied. We mention here the papers Berend-Frougny [6] , Frougny [12, 13], Frougny-Solomyak [14, 15] and Loraud [25] and refer to the references given there. Another kind of number systems which admit the representation of a set which is different from N are the so-called canonical number systems (for short CNS). Since CNS form the main object studied in the present paper we recall their definition (cf. Akiyama-Pethő [2]).

Analysis of linear combination algorithms in cryptography

Several cryptosystems rely on fast calculations of linear combinations in groups. One way to achieve this is to use joint signed binary digit expansions of small “weight.” We study two algorithms, one based on nonadjacent forms of the coefficients of the linear combination, the other based on a certain joint sparse form specifically adapted to this problem. Both methods are sped up using the sliding windows approach combined with precomputed lookup tables. We give explicit and asymptotic results for the number of group operations needed, assuming uniform distribution of the coefficients. Expected values, variances and a central limit theorem are proved using generating functions.Furthermore, we provide a new algorithm that calculates the digits of an optimal expansion of pairs of integers from left to right. This avoids storing the whole expansion, which is needed with the previously known right-to-left methods, and allows an online computation.

On the boundary connectedness of connected tiles

In this paper we consider topological properties of ordinary as well as graph directed iterated function systems. First we give the basic definitions.

Substitutions, Rauzy fractals and tilings

This chapter focuses on multiple tilings associated with substitutive dynamical systems. We recall that a substitutive dynamical system (Xσ, S) is a symbolic dynamical system where the shift S acts on the set Xσ of infinite words having the same language as a given infinite word which is generated by powers of a primitive substitution σ. We restrict to the case where the inflation factor of the substitution σ is a unit Pisot number. With such a substitution σ, we associate a multiple tiling composed of tiles which are given by the unique solution of a set equation expressed in terms of a graph associated with the substitution σ: these tiles are attractors of a graph-directed iterated function system (GIFS). They live in R, where n stands for the cardinality of the alphabet of the substitution. Each of these tiles is compact, it is the closure of its interior, it has non-zero measure and it has a fractal boundary that is also an attractor of a GIFS. These tiles are called central tiles or Rauzy fractals, according to G. Rauzy who introduced them in (Rauzy 1982). Central tiles were first introduced in (Rauzy 1982) for the case of the Tribonacci substitution (1 7→ 12, 2 7→ 13, 3 7→ 1), and then in (Thurston 1989) for the case of the beta-numeration associated with the Tribonacci number (which is the positive root of X − X − X − 1). One motivation for Rauzy’s construction was to exhibit explicit factors of the substitutive dynamical system (Xσ, S) as translations on compact abelian groups, under the hypothesis that σ is a Pisot substitution. By extending the seminal construction in (Rauzy 1982), it has been proved that central tiles can be associated with Pisot substitutions (see for instance (Arnoux and Ito 2001) or (Canterini and Siegel 2001b)) as well as

Fractal properties of number systems

In this paper we study properties of the fundamental domain F of number systems in the n-dimensional real vector space. In particular we investigate the fractal structure of its boundary F. In a first step we give upper and lower bounds for its box counting dimension. Under certain circumstances these bounds are identical and we get an exact value for the box counting dimension. Under additional assumptions we prove that the Hausdorf dimension of F is equal to its box counting dimension. Moreover, we show that the Hausdorf measure is positive and fnite. This is done by applying the theory of graphdirected self similar sets due to Falconer and Bandt. Finally, we discuss the connection to canonical number systems in number felds, and give some numerical examples.

Fractal tiles associated with shift radix systems

Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings. In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials. We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).

Generalized radix representations and dynamical systems III

For r = (r1, . . . , rd) ∈ Rd the map τr : Zd → Zd given by τr(a1, . . . , ad) = (a2, . . . , ad,−br1a1 + · · ·+ rdadc) is called a shift radix system if for each a ∈ Zd there exists an integer k > 0 with τk r (a) = 0. As shown in the first two parts of this series of papers shift radix systems are intimately related to certain well-known notions of number systems like β-expansions and canonical number systems. In the present paper further structural relationships between shift radix systems and canonical number systems are investigated. Among other results we show that canonical number systems related to polynomials

Waring’s problem with digital restrictions

AbstractThe aim of this paper is to consider an analogue of Waring’s problem with digital restrictions. In particular, we prove the following result. Letsq(n) be theq-adic sum of digits function and leth,m be fixed positive integers. Then fors>2k there existsn0∈ℕ such that each integern≥n0 has a representation of the form