arXiv subjects

Jörg M. Thuswaldner

Publications and source records attributed to Jörg M. Thuswaldner.

At least 19 recordsLinked to original sources

arXiv2026

Nonstationary Markov Partitions and Multidimensional Continued Fraction Algorithms

It is well known from results of Sina\uı and Bowen that a hyperbolic toral automorphism admits a Markov partition. Our aim is to generalize this concept to the nonstationary case, i.e., we associate Markov partitions to nonstationary sequences of toral automorphisms. Special emphasis is placed on sequences of toral automorphisms produced by strongly convergent multidimensional continued fraction algorithms. The convergence of the algorithms is expressed in terms of a Pisot type condition which yields hyperbolicity for the nonstationary dynamics with a splitting into two subspaces of dimension 1 and codimension 1, respectively. For a multidimensional continued fraction map, we first consider its natural extension, whose orbits are given by bi-infinite sequences of matrices with determinant $\pm 1$. The Pisot type condition allows us to interpret an orbit of this natural extension as an Anosov mapping family, i.e., as a bi-infinite sequence of toral automorphisms with well-defined stable and unstable manifolds. We prove that this Anosov mapping family admits a bi-infinite sequence of explicit nonstationary Markov partitions. To obtain the atoms of the Markov partitions, a combinatorial structure, expressed in terms of symbolic dynamical systems, namely substitutive and $\mathcal{S}$-adic shifts, has to be superimposed on the Anosov mapping family. In particular, the atoms of the Markov partitions are geometric realizations of $\mathcal{S}$-adic shifts, defined by suspensions of so-called $\mathcal{S}$-adic Rauzy fractals. These Markov partitions then provide a symbolic model as a nonstationary edge shift for the Anosov mapping family. Restacking of the Markov partition yields a renormalization process that allows us to interpret a multidimensional continued fraction algorithm as a sequence of iteratively induced toral rotations.

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arXiv2025

Neighbors of self-affine tiles and Rauzy Fractals

Although the theory of self-affine tiles and the theory of Rauzy fractals are quite different from each other, they have some common features. Both, self-affine tiles and Rauzy fractals have tiling properties and these tiling properties can be checked and described by certain graphs, so-called {\it contact graphs} and {\it neighbor graphs}. The contact graph is often quite easy to construct, but only the neighbor graph contains full information on the overlaps of the tiles in the presumed tiling. In the present paper we establish an algorithm that allows to construct the neighbor graph starting from the contact graph. Such an algorithm is already known in the case of self-affine tiles. In the present paper we give a simplified proof of this algorithm that can be extended to the case of Rauzy fractals. Our algorithms are more efficient than naïve algorithms for the construction of the neighbor graph.

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arXiv2025

Addition Automata and Attractors of Digit Systems Corresponding to Expanding Rational Matrices

Let $A$ be an expanding $2 \times 2$ matrix with rational entries and $\mathbb{Z}^2[A]$ be the smallest $A$-invariant $\mathbb{Z}$-module containing $\mathbb{Z}^2$. Let $\mathcal{D}$ be a finite subset of $\mathbb{Z}^2[A]$ which is a complete residue system of $\mathbb{Z}^2[A]/A\mathbb{Z}^2[A]$. The pair $(A,\mathcal{D})$ is called a {\em digit system} with {\em base} $A$ and {\em digit set} $\mathcal{D}$. It is well known that every vector $x \in \mathbb{Z}^2[A]$ can be written uniquely in the form \[ x = d_0 + Ad_1 + \cdots + A^kd_k + A^{k+1}p, \] with $k\in \mathbb{N}$ minimal, $d_0,\dots,d_k \in \mathcal{D}$, and $p$ taken from a finite set of {\em periodic elements}, the so-called {\em attractor} of $(A,\mathcal{D})$. If $p$ can always be chosen to be $0$ we say that $(A,\mathcal{D})$ has the {\em finiteness property}. In the present paper we introduce finite-state transducer automata which realize the addition of the vectors $\pm(1,0)^\top$ and $\pm(0,1)^\top$ to a given vector $x\in \mathbb{Z}^2[A]$ in a number system $(A,\mathcal{D})$ with collinear digit set. These automata are applied to characterize all pairs $(A,\mathcal{D})$ that have the finiteness property and, more generally, to characterize the attractors of these digit systems.

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arXiv2024

A finiteness condition for complex continued fraction algorithms

It is desirable that a given continued fraction algorithm is simple in the sense that the possible representations can be characterized in an easy way. In this context the so-called finite range condition plays a prominent role. We show that this condition holds for complex $\boldsymbolα$-Hurwitz algorithms with parameters $\boldsymbolα\in\mathbb{Q}^2$. This is equivalent to the existence of certain finite partitions related to these algorithms and lies at the root of explorations into their Diophantine properties. Our result provides a partial answer to a recent question formulated by Lukyanenko and Vandehey.

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arXiv2024

Rational self-affine tiles associated to standard and nonstandard digit systems

We consider digit systems $(A,\mathcal{D})$, where $ A \in \mathbb{Q}^{n\times n}$ is an expanding matrix and the digit set $\mathcal{D}$ is a suitable subset of $\mathbb{Q}^n$. To such a system, we associate a self-affine set $\mathcal{F} = \mathcal{F}(A,\mathcal{D})$ that lives in a certain representation space $\mathbb{K}$. If $A$ is an integer matrix, then $\mathbb{K} = \mathbb{R}^n$, while in the general rational case $\mathbb{K}$ contains an additional solenoidal factor. We give a criterion for $\mathcal{F}$ to have positive Haar measure, i.e., for being a rational self-affine tile. We study topological properties of $\mathcal{F}$ and prove some tiling theorems. Our setting is very general in the sense that we allow $(A,\mathcal{D})$ to be a nonstandard digit system. A standard digit system $(A,\mathcal{D})$ is one in which we require $\mathcal{D}$ to be a complete system of residue class representatives w.r.t. a certain naturally chosen residue class ring. Our tools comprise the Frobenius normal form and character theory of locally compact abelian groups.

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arXiv2024

On self-affine tiles that are homeomorphic to a ball

Let $M$ be a $3\times 3$ integer matrix which is expanding in the sense that each of its eigenvalues is greater than $1$ in modulus and let $\mathcal{D} \subset \mathbb{Z}^3$ be a digit set containing $|\det M|$ elements. Then the unique nonempty compact set $T=T(M,\mathcal{D})$ defined by the set equation $MT=T+\mathcal{D}$ is called an integral self-affine tile if its interior is nonempty. If $\mathcal{D}$ is of the form $\mathcal{D}=\{0,v,\ldots, (|\det M|-1)v\}$ we say that $T$ has a collinear digit set. The present paper is devoted to the topology of integral self-affine tiles with collinear digit sets. In particular, we prove that a large class of these tiles is homeomorphic to a closed $3$-dimensional ball. Moreover, we show that in this case $T$ carries a natural CW complex structure that is defined in terms of the intersections of $T$ with its neighbors in the lattice tiling $\{T+z\,:\, z\in \mathbb{Z}^3\}$ induced by $T$. This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron.

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arXiv2023

Generalizations of Sturmian sequences associated with $N$-continued fraction algorithms

Given a positive integer $N$ and $x$ irrational between zero and one, an $N$-continued fraction expansion of $x$ is defined analogously to the classical continued fraction expansion, but with the numerators being all equal to $N$. Inspired by Sturmian sequences, we introduce the $N$-continued fraction sequences $ω(x,N)$ and $\hatω(x,N)$, which are related to the $N$-continued fraction expansion of $x$. They are infinite words over a two letter alphabet obtained as the limit of a directive sequence of certain substitutions, hence they are $S$-adic sequences. When $N=1$, we are in the case of the classical continued fraction algorithm, and obtain the well-known Sturmian sequences. We show that $ω(x,N)$ and $\hatω(x,N)$ are $C$-balanced for some explicit values of $C$ and compute their factor complexity function. We also obtain uniform word frequencies and deduce unique ergodicity of the associated subshifts. Finally, we provide a Farey-like map for $N$-continued fraction expansions, which provides an additive version of $N$-continued fractions, for which we prove ergodicity and give the invariant measure explicitly.

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arXiv2021

Rational matrix digit systems

Let $A$ be a $d \times d$ matrix with rational entries which has no eigenvalue $λ\in \mathbb{C}$ of absolute value $|λ| < 1$ and let $\mathbb{Z}^d[A]$ be the smallest nontrivial $A$-invariant $\mathbb{Z}$-module. We lay down a theoretical framework for the construction of digit systems $(A, \mathcal{D})$, where $\mathcal{D}\subset \mathbb{Z}^d[A]$ finite, that admit finite expansions of the form \[ \mathbf{x}= \mathbf{d}_0 + A \mathbf{d}_1 + \cdots + A^{\ell-1}\mathbf{d}_{\ell-1} \qquad(\ell\in \mathbb{N},\;\mathbf{d}_0,\ldots,\mathbf{d}_{\ell-1} \in \mathcal{D}) \] for every element $\mathbf{x}\in \mathbb{Z}^d[A]$. We put special emphasis on the explicit computation of small digit sets $\mathcal{D}$ that admit this property for a given matrix $A$, using techniques from matrix theory, convex geometry, and the Smith Normal Form. Moreover, we provide a new proof of general results on this finiteness property and recover analogous finiteness results for digit systems in number fields a unified way.

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arXiv2021

Multidimensional continued fractions and symbolic codings of toral translations

It has been a long standing problem to find good symbolic codings for translations on the $d$-dimensional torus that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties. Inspired by Rauzy's approach we construct such codings in terms of multidimensional continued fraction algorithms that are realized by sequences of substitutions. In particular, given any exponentially convergent continued fraction algorithm, these sequences lead to renormalization schemes which produce symbolic codings of toral translations and bounded remainder sets at all scales in a natural way. The exponential convergence properties of a continued fraction algorithm can be viewed in terms of a Pisot type condition imposed on an attached symbolic dynamical system. Using this fact, our approach provides a systematic way to confirm purely discrete spectrum results for wide classes of symbolic dynamical systems. Indeed, as our examples illustrate, we are able to confirm the Pisot conjecture for many well-known families of sequences of substitutions. These examples comprise classical algorithms like the Jacobi--Perron, Brun, Cassaigne--Selmer, and Arnoux--Rauzy algorithms. As a consequence, we gain symbolic codings of almost all translations of the $2$-dimensional torus having factor complexity $2n+1$ that are balanced for words, which leads to multiscale bounded remainder sets. Using the Brun algorithm, we also give symbolic codings of almost all $3$-dimensional toral translations having multiscale bounded remainder sets.

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arXiv2021

Weyl sums over integers with digital restrictions

We estimate Weyl sums over the integers with sum of binary digits either fixed or restricted by some congruence condition. In our proofs we use ideas that go back to a paper by Banks, Conflitti and the first author (2002). Moreover, we apply the "main conjecture" on the Vinogradov mean value theorem which has been established by Bourgain, Demeter and Guth (2016) as well as by Wooley (2016, 2019). We use our result to give an estimate of the discrepancy of point sets that are defined by the values of polynomials at arguments having the sum of binary digits restricted in different ways.

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arXiv2021

A number system with base $-\frac32$

In the present paper we explore a way to represent numbers with respect to the base $-\frac32$ using the set of digits $\{0,1,2\}$. Although this number system shares several properties with the classical decimal system, it shows remarkable differences and reveals interesting new features. For instance, it is related to the field of $2$-adic numbers, and to some ``fractal'' set that gives rise to a tiling of a non-Euclidean space.

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arXiv2020

$\boldsymbol{S}$-adic sequences. A bridge between dynamics, arithmetic, and geometry

A Sturmian sequence is an infinite nonperiodic string over two letters with minimal subword complexity. In two papers, the first written by Morse and Hedlund in 1940 and the second by Coven and Hedlund in 1973, a surprising correspondence was established between Sturmian sequences on one side and rotations by an irrational number on the unit circle on the other. In 1991 Arnoux and Rauzy observed that an induction process (invented by Rauzy in the late 1970s), related with the classical continued fraction algorithm, can be used to give a very elegant proof of this correspondence. This process, known as the Rauzy induction, extends naturally to interval exchange transformations (this is the setting in which it was first formalized). It has been conjectured since the early 1990s that these correspondences carry over to rotations on higher dimensional tori, generalized continued fraction algorithms, and so-called $S$-adic sequences generated by substitutions. The idea of working towards such a generalization is known as Rauzy's program. Recently Berthé, Steiner, and Thuswaldner made some progress on Rauzy's program and were indeed able to set up the conjectured generalization of the above correspondences. Using a generalization of Rauzy's induction process in which generalized continued fraction algorithms show up, they proved that under certain natural conditions an $S$-adic sequence gives rise to a dynamical system which is measurably conjugate to a rotation on a higher dimensional torus. Moreover, they established a metric theory which shows that counterexamples like the one constructed in 2000 by Cassaigne, Ferenczi, and Zamboni are rare. It is the aim of the present chapter to survey all these ideas and results.

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arXiv2019

The level of distribution of the sum-of-digits function of linear recurrence number systems

Let $G=(G_j)_{j\ge 0}$ be a strictly increasing linear recurrent sequence of integers with $G_0=1$ having characteristic polynomial $X^{d}-a_1X^{d-1}-\cdots-a_{d-1}X-a_d$. It is well known that each positive integer $ν$ can be uniquely represented by the so-called greedy expansion $ν=\varepsilon_0(ν)G_0+\cdots+\varepsilon_\ell(ν)G_\ell$ for $\ell \in \mathbb{N}$ satisfying $G_\ell \le ν< G_{\ell+1}$. Here the digits are defined recursively in a way that $0\le ν- \varepsilon_{\ell}(ν) G_\ell - \cdots - \varepsilon_j(ν) G_j < G_j$ holds for $0 \le j \le \ell$. In the present paper we study the sum-of-digits function $s_G(ν)=\varepsilon_0(ν)+\cdots+\varepsilon_\ell(ν)$ under certain natural assumptions on the sequence $G$. In particular, we determine its level of distribution $x^{\vartheta}$. To be more precise, we show that for $r,s\in\mathbb{N}$ with $\gcd(a_1+\cdots+a_d-1,s)=1$ we have for each $x\ge 1$ and all $A,\varepsilon\in\mathbb{R}_{>0}$ that \[ \sum_{q<x^{\vartheta-\varepsilon}}\max_{z<x}\max_{1\leq h\leq q} \lvert\sum_{\substack{k<z,s_G(k)\equiv r\bmod s\\ k\equiv h\bmod q}}1 -\frac1q\sum_{k<z,s_G(k)\equiv r\bmod s}1\rvert \ll x(\log 2x)^{-A}. \] Here $\vartheta=\vartheta(G) \ge \frac12$ can be computed explicitly and we have $\vartheta(G) \to 1$ for $a_1\to\infty$. As an application we show that $\#\{ k\le x \,:\, s_G(k) \equiv r \pmod{s}, \; k \hbox{ has at most two prime factors} \} \gg x/\log x $ provided that the coefficient $a_1$ is not too small. Moreover, using Bombieri's sieve an "almost prime number theorem" for $s_G$ follows from our result. Our work extends earlier results on the classical $q$-ary sum-of-digits function obtained by Fouvry and Mauduit.

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arXiv2019

Number systems over general orders

Let $\mathcal{O}$ be an order, that is a commutative ring with $1$ whose additive structure is a free $\mathbb{Z}$-module of finite rank. A generalized number system (GNS for short) over $\mathcal{O}$ is a pair $(p,\mathcal{D} )$ where $p\in\mathcal{O}[x]$ is monic with constant term $p(0)$ not a zero divisor of $\mathcal{O}$, and where $\mathcal{D}$ is a complete residue system modulo $p(0)$ in $\mathcal{O}$ containing $0$. We say that $(p,\mathcal{D})$ is a GNS over $\mathcal{O}$ with the finiteness property if all elements of $\mathcal{O}[x]/(p)$ have a representative in $\mathcal{D}[x]$ (the polynomials with coefficients in $\mathcal{D}$). Our purpose is to extend several of the results from a previous paper of Pethő and Thuswaldner, where GNS over orders of number fields were considered. We prove that it is algorithmically decidable whether or not for a given order $\mathcal{O}$ and GNS $(p,\mathcal{D})$ over $\mathcal{O}$, the pair $(p,\mathcal{D})$ admits the finiteness property. This is closely related to work of Vince on matrix number systems. Let $\mathcal{F}$ be a fundamental domain for $\mathcal{O} \!\otimes_{\mathbb{Z}}\! \mathbb{R}/\mathcal{O}$ and $p\in \mathcal{O}[X]$ a monic polynomial. For $α\in\mathcal{O}$, define $p_α(x):=p(x+α)$ and $\mathcal{D}_{\mathcal{F} ,p(α)}:= p(α)\mathcal{F}\cap\mathcal{O}$. Under mild conditions we show that the pairs $(p_α,\mathcal{D}_{\mathcal{F},p(α)}\,)$ are GNS over $\mathcal{O}$ with finiteness property provided $α\in\mathcal{O}$ in some sense approximates a sufficiently large positive rational integer. In the opposite direction we prove under different conditions that $(p_{-m},\mathcal{D}_{\mathcal{F} ,p(-m)}\,)$ does not have the finiteness property for each large enough positive rational integer $m$.

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arXiv2017

Discrepancy bounds for $\boldsymbolβ$-adic Halton sequences

Van der Corput and Halton sequences are well-known low-discrepancy sequences. Almost twenty years ago Ninomiya defined analogues of van der Corput sequences for $β$-numeration and proved that they also form low-discrepancy sequences if $β$ is a Pisot number. Only very recently Robert Tichy and his co-authors succeeded in proving that $\boldsymbolβ$-adic Halton sequences are equidistributed for certain parameters $\boldsymbolβ=(β_1,\ldots,β_s)$ using methods from ergodic theory. In the present paper we continue this research and give discrepancy estimates for $\boldsymbolβ$-adic Halton sequences for which the components $β_i$ are $m$-bonacci numbers. Our methods are quite different and use dynamical and geometric properties of Rauzy fractals that allow to relate $\boldsymbolβ$-adic Halton sequences to rotations on high dimensional tori. The discrepancies of these rotations can then be estimated by classical methods relying on W.~M.~Schmidt's Subspace Theorem.

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arXiv2016

Rational digit systems over finite fields and Christol's Theorem

Let $P, Q\in \mathbb{F}_q[X]\setminus\{0\}$ be two coprime polynomials over the finite field $\mathbb{F}_q$ with $\operatorname{deg}{P} > \operatorname{deg}{Q}$. We represent each polynomial $w$ over $\mathbb{F}_q$ by \[w=\sum_{i=0}^k\frac{s_i}{Q}{\left(\frac{P}{Q}\right)}^i\] using a rational base $P/Q$ and digits $s_i\in\mathbb{F}_q[X]$ satisfying $\operatorname{deg}{s_i} < \operatorname{deg}{P}$. Digit expansions of this type are also defined for formal Laurent series over $\mathbb{F}_q$. We prove uniqueness and automatic properties of these expansions. Although the $ω$-language of the possible digit strings is not regular, we are able to characterize the digit expansions of algebraic elements. In particular, we give a version of Christol's Theorem by showing that the digit string of the digit expansion of a formal Laurent series is automatic if and only if the series is algebraic over $\mathbb{F}_q[X]$. Finally, we study relations between digit expansions of formal Laurent series and a finite fields version of Mahler's $3/2$-problem.

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arXiv2015

Self-affine Manifolds

This paper studies closed 3-manifolds which are the attractors of a system of finitely many affine contractions that tile $\mathbb{R}^3$. Such attractors are called self-affine tiles. Effective characterization and recognition theorems for these 3-manifolds as well as theoretical generalizations of these results to higher dimensions are established. The methods developed build a bridge linking geometric topology with iterated function systems and their attractors. A method to model self-affine tiles by simple iterative systems is developed in order to study their topology. The model is functorial in the sense that there is an easily computable map that induces isomorphisms between the natural subdivisions of the attractor of the model and the self-affine tile. It has many beneficial qualities including ease of computation allowing one to determine topological properties of the attractor of the model such as connectedness and whether it is a manifold. The induced map between the attractor of the model and the self-affine tile is a quotient map and can be checked in certain cases to be monotone or cell-like. Deep theorems from geometric topology are applied to characterize and develop algorithms to recognize when a self-affine tile is a topological or generalized manifold in all dimensions. These new tools are used to check that several self-affine tiles in the literature are 3-balls. An example of a wild 3-dimensional self-affine tile is given whose boundary is a topological 2-sphere but which is not itself a 3-ball. The paper describes how any 3-dimensional handlebody can be given the structure of a self-affine 3-manifold. It is conjectured that every self-affine tile which is a manifold is a handlebody.

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arXiv2014

Comments on the height reducing property II

A complex number $α$ is said to satisfy the height reducing property if there is a finite set $F\subset \mathbb{Z}$ such that $\mathbb{Z}[α]=F[α]$, where $\mathbb{Z}$ is the ring of the rational integers. It is easy to see that $α$ is an algebraic number when it satisfies the height reducing property. We prove the relation $\operatorname{Card}(F)\geq \max\{2,\left\vert M_α(0)\right\vert \},$ where $M_α$ is the minimal polynomial of $α$ over the field of the rational numbers, and discuss the related optimal cases, for some classes of algebraic numbers $α$. In addition, we show that there is an algorithm to determine the minimal height polynomial of a given algebraic number, provided it has no conjugate of modulus one.

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