Canonical Number Systems Research Articles

Cubic CNS polynomials, notes on a conjecture of W.J. Gilbert

A conjecture of W.J. Gilbert's on canonical number systems which are defined by cubic polynomials is partially proved, and it is shown that the conjecture is not complete. Applications to power integral bases of simplest and pure cubic number fields are given thereby extending results of S. Körmendi.

Digit systems in polynomial rings over finite fields

In the present paper we define a new kind of digit system in polynomial rings over finite fields. Let F be a finite field and p(x,y)=∑ b j(x)y j∈ F[x,y] be a polynomial in two indeterminates. Furthermore, let N={p∈ F[x] : deg p(x)< deg b 0(x)} . Then (p(x,y), N) is a digit system with base p( x, y) if each element q of the quotient F[x,y]/p(x,y) F[x,y] has a finite representation of the shape q=d 0(x)+d 1(x)y+⋯+d h(x)y h with d i(x)∈ N (0⩽ i⩽ h). We give a complete characterization of all polynomials p( x, y) which give rise to such a digit system and establish some arithmetical properties of the set of all such polynomials. There are striking analogies of these digit systems to the well-known canonical number systems defined in Z[x]/P(x) Z[x] for a polynomial P(x)∈ Z[x] .

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 canonical number systems

Let P(x)=p dx d+⋯+p 0∈ Z[x] be such that d⩾1,p d=1,p 0⩾2 and N={0,1,…,p 0−1} . We are proving in this note a new criterion for the pair {P(x), N} to be a canonical number system. This enables us to prove that if p 2,…,p d−1,∑ i=1 dp i⩾0 and p 0>2∑ i=1 d|p i|, then {P(x), N} is a canonical number system.

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.

Asymptotic Normality of b-Additive Functions on Polynomial Sequences in the Gaussian Number Field

We consider the asymptotic behavior of b-additive functions f with respect to a base b of a canonical number system in the Gaussian number field. In particular, we get a normal limit law for f(P(z)) where P(z) is a polynomial with integer coefficients. Our methods are exponential sums over the Gaussian number field as well as certain results from the theory of uniform distribution.

The complex sum of digits function and primes

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 for ν b . In all proofs the equidistribution of ν b in residue classes plays a crucial rôle. Starting from this fact we use sieve methods and a version of the model of Kubilius to prove our results.

The Moments of the Sum-Of-Digits Function in Number Fields

AbstractWe consider the asymptotic behavior of the moments of the sum-of-digits function of canonical number systems in number fields. Using Delange’s method we obtain the main term and smaller order terms which contain periodic fluctuations.

The Sum of Digits Function in Number Fields: Distribution in Residue Classes

Canonical number systems are the natural generalization ofq-adic number systems to number fields. Such number systems admit a certain representation of each algebraic integer of a given number field with respect to the powers of a given base numberb. The aim of this paper is to study the sum of digits functionνb(γ) of these number systems. In particular, we prove, thatνb(γ) is equidistributed in residue classes modulomin any ideal s. Furthermore, we investigate the sum of digits function over sumsets.

The Sum of Digits Function In Number Fields

In this paper we are concerned with the asymptotic description of the sum of digits function of canonical number systems in number fields. We derive an asymptotic formula for its summatory function whose second term contains a periodic fluctuation. In imaginary quadratic fields this function is uniformly distributed with respect to the argument. 1991 Mathematics Subject Classification 11R04, 11H06.

Self-Similar Lattice Tilings

We study the general question of the existence of self-similar lattice tilings of Euclidean space. A necessary and sufficient geometric condition on the growth of the boundary of approximate tiles is reduced to a problem in Fourier analysis that is shown to have an elegant simple solution in dimension one. In dimension two we further prove the existence of connected self-similar lattice tilings for parabolic and elliptic dilations. These results apply to produce Haar wavelet bases and certain canonical number systems.

Canonical number systems in algebraic number fields

Canonical number systems in algebraic number fields

Canonical number systems in imaginary quadratic fields

Canonical number systems in imaginary quadratic fields