Canonical Number Systems Research Articles

Generalized Middle-Square Method

In this paper, we generalize John von Neumann’s Middle-Square Method (MSM) to canonical number systems (CNS). Additionally, we present some observations and statistical tests of the sequences generated by the described generators.

Interval-filling sequences of order $N$ and a representation of real numbers in canonical number systems

Interval-filling sequences of order $N$ and a representation of real numbers in canonical number systems

Integral domains with canonical number systems

Integral domains with canonical number systems

"Exotic" binary number systems for rings of Gauss and Eisenstein integers

The paper considers nonstandard binary number systems for rings of Gauss and Eisenstein integers. The principal difference ("exoticism") of such number systems from the canonical number systems introduced by I. Katai for quadratic fields is that as a binary "digital alphabet", it uses a two-element set that does not contain a numeric zero. The paper also synthesizes algorithms for the representation of numbers in the considered number system and characterizes the possibility of an efficient implementation of arithmetic operations.

Ternary number systems in finite fields

The work continues the author's previous study of positional number systems in finite fields. The paper considers ternary number systems and arithmetic operations algorithms for the representation of elements of finite fields in the so-called ternary reduced number systems, which are reductions of the canonical number systems when mapping the corresponding ring of integers of a quadratic field into some prime field. A classification of finite fields in which such number systems exist is given. It is proved that the reduced ternary number systems exist for most finite prime fields.

Calculation of Fourier-Galois transforms in reduced binary number systems

The paper proposes a new method for calculating Fourier-Galois transforms (number-theoretical transforms), which are a modular analog of the discrete Fourier transform. A number of specific problems related to the calculation of transforms in a finite field can be solved by representing the elements of these fields in “exotic” number systems, which are reductions of the canonical number systems proposed by I. Katai when mapping the corresponding ring of an integer quadratic field into a field of the prime residue classes modulo. The case of binary reduced number systems is studied in detail. It is proved that such number systems exist for any prime number.

An infinite family of multiplicatively independent bases of number systems in cyclotomic number fields

Let $\zeta_k$ be a $k$-th primitive root of unity, $m\geq\phi(k)+1$ an integer and $\Phi_k(X)\in\mathbb Z [X]$ the $k$-th cyclotomic polynomial. In this paper we show that the pair $(-m+\zeta_k,\mathcal N)$ is a canonical number system, with $\mathcal N=\{0,1,\dots,|\Phi_k(m)|\}$. Moreover we also discuss whether the two bases $-m+\zeta_k$ and $-n+\zeta_k$ are multiplicatively independent for positive integers $m$ and $n$ and $k$ fixed.

On multiplicatively independent bases in cyclotomic number fields

Recently the authors [12] showed that the algebraic integers of the form $${-m+\zeta_k}$$ are bases of a canonical number system of $${\mathbb {Z}[{\zeta_k]}}$$ provided $${m \geqq \phi(k)+1}$$ , where $${\zeta_k}$$ denotes a k-th primitive root of unity and $${\phi}$$ is Euler’s totient function. In this paper we are interested in the questions whether two bases $${-m+\zeta_k}$$ and $${-n+\zeta_k}$$ are multiplicatively independent. We show the multiplicative independence in case that 0 1.

Rational self-affine tiles

An integral self-affine tile is the solution of a set equation A T = ⋃ d ∈ D ( T + d ) \mathbf {A} \mathcal {T} = \bigcup _{d \in \mathcal {D}} (\mathcal {T} + d) , where A \mathbf {A} is an n × n n \times n integer matrix and D \mathcal {D} is a finite subset of Z n \mathbb {Z}^n . In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices A ∈ Q n × n \mathbf {A} \in \mathbb {Q}^{n \times n} . We define rational self-affine tiles as compact subsets of the open subring R n × ∏ p K p \mathbb {R}^n\times \prod _\mathfrak {p} K_\mathfrak {p} of the adèle ring A K \mathbb {A}_K , where the factors of the (finite) product are certain p \mathfrak {p} -adic completions of a number field K K that is defined in terms of the characteristic polynomial of A \mathbf {A} . Employing methods from classical algebraic number theory, Fourier analysis in number fields, and results on zero sets of transfer operators, we establish a general tiling theorem for these tiles. We also associate a second kind of tile with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with R n × ∏ p { 0 } ≃ R n \mathbb {R}^n \times \prod _\mathfrak {p} \{0\} \simeq \mathbb {R}^n . Although these intersection tiles have a complicated structure and are no longer self-affine, we are able to prove a tiling theorem for these tiles as well. For particular choices of the digit set D \mathcal {D} , intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. This enables us to gain new results for tilings associated with numeration systems.

Power integral bases for certain pure sextic fields

In this paper, we characterize whether the pure sextic fields [Formula: see text] with square-free integers m ≢ ±1( Mod 9) have power integral bases or do not; if m ≡ 2, 3 (Mod 4), then [Formula: see text] have power integral bases. We prove this by determining relative integral bases of such fields with respect to their cubic and quadratic subfields. Based on the works of Kovács and Pethő, several examples on application of monogenic fields to CNS (Canonical Number System) are shown.

Digit systems over commutative rings

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be a polynomial. In the present paper we consider digit representations in the residue class ring [Formula: see text]. In particular, we are interested in the question whether each [Formula: see text] can be represented modulo P in the form e0+ e1x + ⋯ + ehxh, where the [Formula: see text] are taken from a fixed finite set of digits. This general concept generalizes both canonical number systems and digit systems over finite fields. Due to the fact that we do not assume that 0 is an element of the digit set and that P need not be monic, several new phenomena occur in this context.

Fractal discrete cosine transform on pre-fractal domains associated with fundamental domains of canonical number systems

Introduce an analog cosine transform on the pre-fractal region. From fractal discrete Fourier transform output discrete cosine transform on two-dimensional domains associated with the fundamental domains of number systems in imaginary quadratic rings. It is shown that this transformation is orthogonal.

Discrete orthogonal transforms on fundamental domains of canonical number systems

Advertised methods for the synthesis of discrete orthogonal transformations on two-dimensional fields associated with the fundamental domains of number systems in quadratic rings.

The moments of b-additive functions in canonical number systems

The moments of b-additive functions in canonical number systems

The sum-of-digits function of canonical number systems: Distribution in residue classes

Abstract In the present paper we investigate the sum-of-digits function for canonical number systems. We are interested in its re-partition in arithmetic progressions and similar results for sum sets. The original problem goes back to Gelfond, who proved the independence of the distribution of the digits and their sum of digits. The present paper extends results by Thuswaldner and Mauduit and Sarkozy to this kind of numeration systems.

ROOTS OF POLYNOMIALS WITH DOMINANT TERM

We characterize all algebraic numbers which are roots of integer polynomials with a coefficient whose modulus is greater than or equal to the sum of moduli of all the remaining coefficients. It turns out that these numbers are zero, roots of unity and those algebraic numbers β whose conjugates over ℚ (including β itself) do not lie on the circle |z| = 1. We also describe all algebraic integers with norm B which are roots of an integer polynomial with constant coefficient B and the sum of moduli of all other coefficients at most |B|. In contrast to the above, the set of such algebraic integers is "quite small". These results are motivated by a recent paper of Frougny and Steiner on the so-called minimal weight β-expansions and are also related to some work on canonical number systems and tilings.

Asymptotic normality of additive functions on polynomial sequences in canonical number systems

The objective of this paper is the study of functions which only act on the digits of an expansion. In particular, we are interested in the asymptotic distribution of the values of these functions. The presented result is an extension and generalization of a result of Bassily and Kátai to number systems defined in a quotient ring of the ring of polynomials over the integers.

Parallel implementations of Brunotte’s algorithm

In this paper the author make a comprehensive comparison of different parallelizations of a sequential number theoretic algorithm having large memory requirements. Brunotte’s algorithm is one of the currently known best methods for the decision of the canonical number system (or more generally shift radix system) property. Still, it can be very space-consuming in some cases. Pushing the algorithm to its limits may hopefully shed light on mathematical patterns that would otherwise not be discernible. The algorithm contains many n -dimensional vector operations and set operations like insert, find, clear, etc. The parallel algorithms encounter two difference kinds of concurrency problems. First, they need computationally intensive arithmetic vector operations, second, the set implementations require a huge amount of memory and general purpose processors. The algorithms described in this article are basically designed for two platforms. The first platform is a generic symmetric multiprocessing (SMP) architecture without any vector processor extension, the second is the Cell Broadband Engine. The SMP platforms have several general purpose processors in contrast with the Cell Broadband Engine where the processors have Synergistic vector processors.

INTERIOR COMPONENTS OF A TILE ASSOCIATED TO A QUADRATIC CANONICAL NUMBER SYSTEM — PART II

Let [Formula: see text] be a root of the polynomial p(x) = x2+ 4x + 5. It is well-known that the pair (α, {0, 1, 2, 3, 4}) forms a canonical number system, i.e., that each γ ∈ ℤ[α] admits a finite representation of the shape γ = a0+ a1α + ⋯ + aℓαℓwith ai∈ {0, 1, 2, 3, 4}. The set [Formula: see text] of points with integer part 0 in this number system [Formula: see text] is called the fundamental domain of this canonical number system. It is a plane continuum with nonempty interior which induces a tiling of ℂ. Moreover, it has a disconnected interior [Formula: see text]. In the first paper of this series we described the closures C0, C1, C2and C3of the four largest components of [Formula: see text] as attractors of graph-directed self-similar sets. Each of these four sets is a translation of C0. We conjectured that the closures of the other components are images of C0by similarity transformations. In this article we prove this conjecture. Moreover, we provide a graph from which the suitable similarities can be read off.

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).