arXiv subjects

Wolfgang Steiner

Publications and source records attributed to Wolfgang Steiner.

52 records · Page 3Linked to original sources

arXiv2012

Patterns in rational base number systems

Number systems with a rational number $a/b > 1$ as base have gained interest in recent years. In particular, relations to Mahler's 3/2-problem as well as the Josephus problem have been established. In the present paper we show that the patterns of digits in the representations of positive integers in such a number system are uniformly distributed. We study the sum-of-digits function of number systems with rational base $a/b$ and use representations w.r.t. this base to construct normal numbers in base $a$ in the spirit of Champernowne. The main challenge in our proofs comes from the fact that the language of the representations of integers in these number systems is not context-free. The intricacy of this language makes it impossible to prove our results along classical lines. In particular, we use self-affine tiles that are defined in certain subrings of the adéle ring $\mathbb{A}_\mathbb{Q}$ and Fourier analysis in $\mathbb{A}_\mathbb{Q}$. With help of these tools we are able to reformulate our results as estimation problems for character sums.

math.NT↗

arXiv2011

Digital expansions with negative real bases

Similarly to Parry's characterization of $β$-expansions of real numbers in real bases $β> 1$, Ito and Sadahiro characterized digital expansions in negative bases, by the expansions of the endpoints of the fundamental interval. Parry also described the possible expansions of 1 in base $β> 1$. In the same vein, we characterize the sequences that occur as $(-β)$-expansion of $\frac{-β}{β+1}$ for some $β> 1$. These sequences also describe the itineraries of 1 by linear mod one transformations with negative slope.

math.DS↗

arXiv2011

On the Delone property of (-β)-integers

The (-β)-integers are natural generalisations of the β-integers, and thus of the integers, for negative real bases. They can be described by infinite words which are fixed points of anti-morphisms. We show that they are not necessarily uniformly discrete and relatively dense in the real numbers.

cs.FL↗

arXiv2011

Dynamical properties of the negative beta transformation

We analyse dynamical properties of the negative beta transformation, which has been studied recently by Ito and Sadahiro. Contrary to the classical beta transformation, the density of the absolutely continuous invariant measure of the negative beta transformation may be zero on certain intervals. By investigating this property in detail, we prove that the $(-β)$-transformation is exact for all $β>1$, confirming a conjecture of Góra, and intrinsic, which completes a study of Faller. We also show that the limit behaviour of the $(-β)$-expansion of 1 when $β$ tends to 1 is related to the Thue-Morse sequence. A consequence of the exactness is that every Yrrap number, which is a $β>1$ such that the $(-β)$-expansion of 1 is eventually periodic, is a Perron number. This extends a well-known property of Parry numbers. However, the set of Parry numbers is different from the set of Yrrap numbers.

math.DS↗

arXiv2011

On the structure of ($-β$)-integers

The $(-β)$-integers are natural generalisations of the $β$-integers, and thus of the integers, for negative real bases. When $β$ is the analogue of a Parry number, we describe the structure of the set of $(-β)$-integers by a fixed point of an anti-morphism.

math.NT↗

arXiv2011

Redundancy of minimal weight expansions in Pisot bases

Motivated by multiplication algorithms based on redundant number representations, we study representations of an integer $n$ as a sum $n=\sum_k ε_k U_k$, where the digits $ε_k$ are taken from a finite alphabet $Σ$ and $(U_k)_k$ is a linear recurrent sequence of Pisot type with $U_0=1$. The most prominent example of a base sequence $(U_k)_k$ is the sequence of Fibonacci numbers. We prove that the representations of minimal weight $\sum_k|ε_k|$ are recognised by a finite automaton and obtain an asymptotic formula for the average number of representations of minimal weight. Furthermore, we relate the maximal order of magnitude of the number of representations of a given integer to the joint spectral radius of a certain set of matrices.

math.NT↗

arXiv2010

Fractal tiles associated with shift radix systems

Shift radix systems form a collection of dynamical systems depending on a parameter $\mathbf{r}$ 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 $\mathbf{r}$ 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 $\mathbf{r}$ 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).

math.DS↗

arXiv2010

Beta-expansions, natural extensions and multiple tilings associated with Pisot units

From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $β$ and the greedy $β$-transformation. In this paper, we consider different transformations generating expansions in base $β$, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $β$-expansions. Remarkably, the symmetric $β$-transformation does not satisfy this condition when $β$ is the smallest Pisot number or the Tribonacci number. This means that the Pisot conjecture on tilings cannot be extended to the symmetric $β$-transformation. Closely related to these (multiple) tilings are natural extensions of the transformations, which have many nice properties: they are invariant under the Lebesgue measure; under certain conditions, they provide Markov partitions of the torus; they characterize the numbers with purely periodic expansion, and they allow determining any digit in an expansion without knowing the other digits.

math.DS↗

arXiv2010

Regularities of the distribution of abstract van der Corput sequences

Similarly to $β$-adic van der Corput sequences, abstract van der Corput sequences can be defined for abstract numeration systems. Under some assumptions, these sequences are low discrepancy sequences. The discrepancy function is computed explicitely, and a characterization of bounded remainder sets of the form $[0,y)$ is provided.

math.NT↗

arXiv2010

Similar dissection of sets

In 1994, Martin Gardner stated a set of questions concerning the dissection of a square or an equilateral triangle in three similar parts. Meanwhile, Gardner's questions have been generalized and some of them are already solved. In the present paper, we solve more of his questions and treat them in a much more general context. Let $D\subset \mathbb{R}^d$ be a given set and let $f_1,...,f_k$ be injective continuous mappings. Does there exist a set $X$ such that $D = X \cup f_1(X) \cup ... \cup f_k(X)$ is satisfied with a non-overlapping union? We prove that such a set $X$ exists for certain choices of $D$ and $\{f_1,...,f_k\}$. The solutions $X$ often turn out to be attractors of iterated function systems with condensation in the sense of Barnsley. Coming back to Gardner's setting, we use our theory to prove that an equilateral triangle can be dissected in three similar copies whose areas have ratio $1:1:a$ for $a \ge (3+\sqrt{5})/2$.

math.MG↗

arXiv2010

Rational numbers with purely periodic $β$-expansion

We study real numbers $β$ with the curious property that the $β$-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the number field they generate. We complete known results due to Akiyama to characterize algebraic numbers of degree 3 that enjoy this property. This extends results previously obtained in the case of degree 2 by Schmidt, Hama and Imahashi. Let $γ(β)$ denote the supremum of the real numbers $c$ in $(0,1)$ such that all positive rational numbers less than $c$ have a purely periodic $β$-expansion. We prove that $γ(β)$ is irrational for a class of cubic Pisot units that contains the smallest Pisot number $η$. This result is motivated by the observation of Akiyama and Scheicher that $γ(η)=0.666 666 666 086 ...$ is surprisingly close to 2/3.

math.NT↗

arXiv2009

Minimal weight expansions in Pisot bases

For applications to cryptography, it is important to represent numbers with a small number of non-zero digits (Hamming weight) or with small absolute sum of digits. The problem of finding representations with minimal weight has been solved for integer bases, e.g. by the non-adjacent form in base~2. In this paper, we consider numeration systems with respect to real bases $β$ which are Pisot numbers and prove that the expansions with minimal absolute sum of digits are recognizable by finite automata. When $β$ is the Golden Ratio, the Tribonacci number or the smallest Pisot number, we determine expansions with minimal number of digits $\pm1$ and give explicitely the finite automata recognizing all these expansions. The average weight is lower than for the non-adjacent form.

cs.DM↗

arXiv2008

Periodicity of certain piecewise affine planar maps

We determine periodic and aperiodic points of certain piecewise affine maps in the Euclidean plane. Using these maps, we prove for $λ\in\{\frac{\pm1\pm\sqrt5}2,\pm\sqrt2,\pm\sqrt3\}$ that all integer sequences $(a_k)_{k\in\mathbb Z}$ satisfying $0\le a_{k-1}+λa_k+a_{k+1}<1$ are periodic.

math.DS↗

arXiv2008

Abstract numeration systems on bounded languages and multiplication by a constant

A set of integers is $S$-recognizable in an abstract numeration system $S$ if the language made up of the representations of its elements is accepted by a finite automaton. For abstract numeration systems built over bounded languages with at least three letters, we show that multiplication by an integer $λ\ge2$ does not preserve $S$-recognizability, meaning that there always exists a $S$-recognizable set $X$ such that $λX$ is not $S$-recognizable. The main tool is a bijection between the representation of an integer over a bounded language and its decomposition as a sum of binomial coefficients with certain properties, the so-called combinatorial numeration system.

cs.DM↗

arXiv2008

Metrical theory for $α$-Rosen fractions

The Rosen fractions form an infinite family which generalizes the nearest-integer continued fractions. In this paper we introduce a new class of continued fractions related to the Rosen fractions, the $α$-Rosen fractions. The metrical properties of these $α$-Rosen fractions are studied. We find planar natural extensions for the associated interval maps, and show that these regions are closely related to similar region for the 'classical' Rosen fraction. This allows us to unify and generalize results of diophantine approximation from the literature.

math.NT↗

arXiv2007

Sequences with constant number of return words

An infinite word has the property $R_m$ if every factor has exactly $m$ return words. Vuillon showed that $R_2$ characterizes Sturmian words. We prove that a word satisfies $R_m$ if its complexity function is $(m-1)n+1$ and if it contains no weak bispecial factor. These conditions are necessary for $m=3$, whereas for $m=4$ the complexity function need not be $3n+1$. New examples of words satisfying $R_m$ are given by words related to digital expansions in real bases.

math.CO↗