Klaus Scheicher

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]).

β-Expansions in algebraic function fields over finite fields

The present paper deals with an algebraic function field analogue of @b-expansions of real numbers. It completely characterizes the sets with eventually periodic and finite expansions. These characterizations are unknown in the real case.

On the tractability of the Brownian bridge algorithm

Recent results in the theory of quasi-Monte Carlo methods have shown that the weighted Koksma-Hlawka inequality gives better estimates for the error of quasi-Monte Carlo algorithms. We present a method for finding good weights for several classes of functions and apply it to certain algorithms using the Brownian Bridge construction, which are important for financial applications.

Neighbours of Self-affine Tiles in Lattice Tilings

Let T be a tile of a self-affine lattice tiling. We give an algorithm that allows to determine all neighbours of T in the tiling. This can be used to characterize the sets V L of points, where T meets L other tiles. Our algorithm generalizes an algorithm of the authors which was applicable only to a special class of self-affine lattice tilings. This new algorithm can also be applied to classes containing infinitely many tilings at once. Together with the results in recent papers by Bandt and Wang as well as Akiyama and Thuswaldner it allows to characterize classes of plane tilings which are homeomorphic to a disc. Furthermore, it sheds some light on the relations between different kinds of characterizations of the boundary of T.

Number systems and tilings over Laurent series

Let F be a field and F[x, y] the ring of polynomials in two variables over F. Let f ∈ F[x, y] and consider the residue class ring R := F[x, y]/fF[x, y]. Our first aim is to study digit representations in R, i.e., we ask for which f each element r ∈ R admits a digit representation of the form d0 + d1x + · · · + d`x with digits di ∈ F[y] satisfying degy di < degy f . These digit systems are motivated by the well-known notion of canonical number system. In a next step we enlarge the ring in order to allow for representations including negative powers of the “base” x. More precisely we define and characterize digit representations for the ring F((x−1, y−1))/fF((x−1, y−1)) and give easy to handle criteria for finiteness and periodicity. Finally, we attach fundamental domains to our number systems. The fundamental domain of a number system is the set of all numbers having only negative powers of x in their “x-ary” representation. Interestingly, the fundamental domains of our number systems turn out to be unions of boxes. If we choose F = Fq to be a finite field, these unions become finite.

Complexity and effective dimension of discrete Lévy areas

Discretisation methods to simulate stochastic differential equations belong to the main tools in mathematical finance. For Ito processes, there exist several Euler- or Runge-Kutta-like methods which are analogues of well-known approximation schemes in the nonstochastic case. In the multidimensional case, there appear several difficulties, caused by the mixed second order derivatives. These mixed terms (or more precisely their differences) correspond to special random variables called Levy stochastic area terms. In the present paper, we compare three approximation methods for such random variables with respect to computational complexity and the so-called effective dimension.

Digit systems over commutative rings

Let

Automatic β-expansions of formal Laurent series over finite fields

\E

Dynamical properties of the tent map

be a commutative ring with identity and