Canonical Number Systems Research Articles

The sum of digits of squares in [formula omitted

We consider the complex sum-of-digits function s q for squares with respect to special bases q of a canonical number system in the Gaussian integers Z [ i ] . In particular, we show that the sequence ( α s q ( z 2 ) ) z ∈ Z [ i ] is uniformly distributed modulo 1 if and only if α is irrational. Furthermore we introduce special sets of Gaussian integers (related to Følner sequences) for which we can determine the order of magnitude of the number of integers z for which s q ( z 2 ) lies in a fixed residue class mod m. This extends a recent result of Mauduit and Rivat to Z [ i ] . We also improve an estimate of Gittenberger and Thuswaldner in order to show a local limit theorem for the sum-of-digits function of squares. We can provide asymptotic expansions for # { z ∈ Z [ i ] ∩ D N : s q ( z 2 ) = k } where ( D N ) N ∈ N is a sequence of convex sets.

Number systems and tilings over Laurent series

AbstractLetbe a field and[x,y] the ring of polynomials in two variables over. Letf∈[x,y] and consider the residue class ringR:=[x,y]/f[x,y]. Our first aim is to study digit representations inR, i.e., we ask for whichfeach element ofRadmits a digit representation of the formd0+d1x+ ⋅ ⋅ ⋅ +dℓxℓwith digitsdi∈[y] satisfying degy(di) < degy(f). These digit systems are motivated by the well-known notion of canonical number systems. Next we enlarge the ring in order to allow representations including negative powers of the “base”x. In particular, we define and characterize digit representations for the ringS:=((x−1,y−1))/f((x−1,y−1)) and give easy to handle criteria for finiteness and periodicity of such representations. Finally, we attach fundamental domains to our digit systems. The fundamental domain of a digit system is the set of all elements having only negative powers ofxin their “x-ary” representation. The translates of the fundamental domain induce a tiling ofS. Interestingly, the fundamental domains of our digit systems turn out to be unions of boxes. If we choose=qto be a finite field, these unions become finite.

Generalized radix representations and dynamical systems. IV

For r = ( r 1,…, r d ) ∈ ℝ d the mapping τ r :ℤ d →ℤ d given by τ r ( a 1,…, a d ) = ( a 2, …, a d ,−⌊ r 1 a 1+…+ r d a d ⌋) where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ℤ d there exists an integer k > 0 with τ r k ( a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).

Fundamental group of tiles associated to quadratic canonical number systems

Abstract If A is a 2 × 2 expanding matrix with integral coefficients, and ⊂ ℤ2 a complete set of coset representatives of ℤ2/Aℤ2 with |det(A)| elements, then the set ℐ defined by Aℐ = ℐ + is a self-affine plane tile of ℝ2, provided that its two-dimensional Lebesgue measure is positive. It was shown by Luo and Thuswaldner that the fundamental group of such a tile is either trivial or uncountable. To a quadratic polynomial x 2 + Ax + B, A, B ∈ ℤ such that B ≥ 2 and −1 ≤ A ≤ B, one can attach a tile ℐ. Akiyama and Thuswaldner proved the triviality of the fundamental group of this tile for 2A < B + 3, by showing that a tile of this class is homeomorphic to a closed disk. The case 2A ≥ B + 3 is treated here by using the criterion given by Luo and Thuswaldner.

Interior components of a tile associated to a quadratic canonical number system

Let α = − 2 + −1 be a root of the polynomial p ( x ) = x 2 + 4 x + 5 . It is well known that the pair ( p ( x ) , { 0 , 1 , 2 , 3 , 4 } ) forms a canonical number system, i.e., that each x ∈ Z [ α ] admits a finite representation of the shape x = a 0 + a 1 α + ⋯ + a ℓ α ℓ with a i ∈ { 0 , 1 , 2 , 3 , 4 } . The set T of points with integer part 0 in this number system T : = { ∑ i = 1 ∞ a i α − i , a i ∈ { 0 , 1 , 2 , 3 , 4 } } is called the fundamental domain of this canonical number system. It has been studied extensively in the literature. Up to now it is known that it is a plane continuum with nonempty interior which induces a tiling of the plane. However, its interior is disconnected. In the present paper we describe some of (the closures of) the components of its interior as attractors of graph directed self-similar constructions. The associated graph can also be used in order to determine the Hausdorff dimension of the boundary of these components. Amazingly, this dimension is strictly smaller than the Hausdorff dimension of the boundary of T .

Generating normal numbers over Gaussian integers

In this paper we consider normal numbers over Gaussian integers generated by polynomials. This corresponds to results by Nakai and Shiokawa in the case of real numbers. A generalization of normal numbers to Gaussian integers and their canonical number systems, which were characterized by Katai and Szabo, is given. With help of this we construct normal numbers of the form 0. bf(z1)c bf(z2)c bf(z3)c bf(z4)c bf(z5)c bf(z6)c . . . Here we denote by bxc the expansion of the integer part of x with respect to a given number system in Z[i], by f we denote a polynomial with complex coefficients, and by zi we denote a numbering of the Gaussian integers. We are able to show, that a number, which is constructed in this way, is normal to the given number system.

Reducible cubic CNS polynomials

The concept of a canonical number system can be regarded as a natural generalization of decimal representations of rational integers to elements of residue class rings of polynomial rings. Generators of canonical number systems are CNS polynomials which are known in the linear and quadratic cases, but whose complete description is still open. In the present note reducible CNS polynomials are treated, and the main result is the characterization of reducible cubic CNS polynomials.

Three-dimensional symmetric shift radix systems

Shift radix systems have been introduced by Akiyama et al. as a common generalization of β-expansions and canonical number systems. In the present paper we study a variant of them, so-called symmetric shift radix systems which were introduced recently by Akiyama and Scheicher. In particular, for d ∈ N and r ∈ R let (a = (a1, . . . , ad)) τr : Z d → Z, a 7→ (

Cryptography and canonical number systems in quadratic fields

This paper proposes an encryption method based on representation of messages in the canonical number systems (CNS) in quadratic fields. The essence of the encryption method is conversion of the rep...

Bases of canonical number systems in quartic algebraic number fields

Canonical number systems can be viewed as natural generalizations of radix representations of ordinary integers to algebraic integers. A slightly modified version of an algorithm of B. Kovács and A. Pethő is presented here for the determination of canonical number systems in orders of algebraic number fields. Using this algorithm canonical number systems of some quartic fields are computed.

Generalized radix representations and dynamical systems II

We are concerned with families of dynamical systems which are related to generalized radix representations. The properties of these dynamical systems lead to new results on the characterization of bases of Pisot number systems as well as canonical number systems.

Arcwise connectedness of the boundary of connected self-similar sets

Let Tbe the connected attractor of injective contractions f1,... fmon Rd that satisfy the Open Set Condition. We show that ∂Tis arcwise connected. In particular, the boundary of the Levy dragon and those of the fundamental domains of canonical number systems are arcwise connected.

Generalized radix representations and dynamical systems. I

We are concerned with families of dynamical systems which are related to generalized radix representations. The properties of these dynamical systems lead to new results on the characterization of bases of Pisot number systems as well as canonical number systems.

Vertices of self-similar tiles

The set $V_n$ of $n$-vertices of a tile $T$ in $\R^d$ is the common intersection of $T$ with at least $n$ of its neighbors in a tiling determined by $T$. Motivated by the recent interest in the topological structure as well as the associated canonical number systems of self-similar tiles, we study the structure of $V_n$ for general and strictly self-similar tiles. We show that if $T$ is a general self-similar tile in $\R^2$ whose interior consists of finitely many components, then any tile in any self-similar tiling generated by $T$ has a finite number of vertices. This work is also motivated by the efforts to understand the structure of the well-known L\'evy dragon. In the case $T$ is a strictly self-similar tile or multitile in $\R^d$, we describe a method to compute the Hausdorff and box dimensions of $V_n$. By applying this method, we obtain the dimensions of the set of $n$-vertices of the L\'evy dragon for all $n\ge 1$.

The topological structure of fractal tilings generated by quadratic number systems

Let α be a root of an irreducible quadratic polynomial x 2 + Ax + B with integer coefficients A, B and assume that α forms a canonical number system, i.e., each x ∈ ℤ[ α] admits a representation of the shape x = a 0 + a 1 α + ⋯ + a h α h , with a i ∈ {0, 1,…,| B| - 1}. It is possible to associate a tiling to such a number system in a natural way. If 2 A < B + 3, then we show that the fractal boundary of the tiles of this tiling is a simple closed curve and its interior is connected. Furthermore, the exact set equation for the boundary of a tile is given. If 2 A ≥ B + 3, then the topological structure of the tiles is quite involved. In this case, we prove that the interior of a tile is disconnected. Furthermore, we are able to construct finite labelled directed graphs which allow to determine the set of “neighbours” of a given tile T, i.e., the set of all tiles which have nonempty intersection with T. In a next step, we give the structure of the set of points, in which T coincides with L other tiles. In this paper, we use two different approaches: geometry of numbers and finite automata theory. Each of these approaches has its advantages and emphasizes different properities of the tiling. In particular, the conjecture in [1], that for A ≠ 0 and 2 A < B + 3 there exist exactly six points where T coincides with two other tiles, is solved in these two ways in Theorems 6.6 and 10.1.

Connectedness of number theoretical tilings

Let T=T(A,D) be a self-affine tile in \reals^n defined by an integral expanding matrix A and a digit set D. In connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers \0,1,..., |det(A)|-1\. It is shown that in \reals^3 and \reals^4, for any integral expanding matrix A, T(A,D) is connected. We also study the connectedness of Pisot dual tilings which play an important role in the study of β -expansion, substitution and symbolic dynamical system. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. This is naturally expected since the digit set consists of consecutive integers as above. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, a complete classification of the β -expansion of 1 for quartic Pisot units is given.

A TECHNIQUE IN THE TOPOLOGY OF CONNECTED SELF-SIMILAR TILES

Not much is known about the topological structure of a connected self-similar tile whose interior is disconnected, and even less is understood if the interior consists of infinitely many components. We introduce a technique to show that for a large class of self-similar tiles in ℝ2, the closure of each component of the interior is homeomorphic to a disk. This allows us to prove such a result for the Eisenstein set, the fundamental domain of a well-known quadratic canonical number system, and some other well-known fractal tiles.

On the characterization of canonical number systems

On the characterization of canonical number systems

On the connectedness of self-affine attractors

Let T = T(A, D) be a self-affine attractor in \( \mathbb{R}^n \) defined by an integral expanding matrix A and a digit set D. In the first part of this paper, in connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers \( \{0, 1,\ldots, |\det(A)| - 1\} \) . It is shown that in \( \mathbb{R}^3 \) and \( \mathbb{R}^4 \) , for any integral expanding matrix A, T(A, D) is connected.

New criteria for canonical number systems

Let P (x) = x + pd−1xd−1 + · · · + p0 be an expanding monic polynomial with integer coefficients. If each element of Z[x]/P (x)Z[x] has a polynomial representative with coefficients in [0, |p0| − 1] then P (x) is called a canonical number system generating polynomial, or a CNS polynomial in short. A method due to Hollander [6] is employed to study CNS polynomials. Several new criteria for canonical number system generating polynomials are given and a conjecture of S.Akiyama & A.Pethő [3] is proved. The known results, especially an algorithm of H. Brunotte’s in [4] and a recent work of K. Scheicher & J.M.Thuswaldner [15], can be derived by this new method in a simpler way.