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A dynamical view of Tijdeman's solution of the chairman assignment problem

Valérie Berthé, Olivier Carton, Nicolas Chevallier, Wolfgang Steiner, Reem Yassawi

TL;DR

This work develops a dynamical framework for Tijdeman's low-discrepancy chairman-sequence construction by showing they arise as bounded natural codings of totally irrational toral translations $T_{\boldsymbol{\alpha}}$ with respect to polytopal partitions. It proves these systems are minimal, uniquely ergodic, and have purely discrete spectrum, and it establishes polynomial (in fact polyhedral) factor complexity growth. The paper compares two constructions—hypercubic billiard codings and Tijdeman's refined method—and shows that the latter achieves discrepancy near the optimal bound $D_d = 1 - 1/(2d-2)$ via a cut-and-project model with a polytopal window. Together, these results connect discrepancy theory, symbolic dynamics, and aperiodic order (model sets) to provide a rigorous, computable description of Tijdeman sequences in a dynamical systems framework, with explicit complexity bounds and spectral properties.

Abstract

In 1980, R. Tijdeman provided an on-line algorithm that generates sequences over a finite alphabet with minimal discrepancy, that is, such that the occurrence of each letter optimally tracks its frequency. In this article, we define discrete dynamical systems generating these sequences. The dynamical systems are defined as exchanges of polytopal pieces, yielding cut and project schemes, and they code tilings of the line whose sets of vertices form model sets. We prove that these sequences of low discrepancy are natural codings of toral translations with respect to polytopal atoms, and that they generate a minimal and uniquely ergodic subshift with purely discrete spectrum. Finally, we show that the factor complexity of these sequences is of polynomial growth order $n^{d-1}$, where $d$ is the cardinality of the alphabet.

A dynamical view of Tijdeman's solution of the chairman assignment problem

TL;DR

This work develops a dynamical framework for Tijdeman's low-discrepancy chairman-sequence construction by showing they arise as bounded natural codings of totally irrational toral translations with respect to polytopal partitions. It proves these systems are minimal, uniquely ergodic, and have purely discrete spectrum, and it establishes polynomial (in fact polyhedral) factor complexity growth. The paper compares two constructions—hypercubic billiard codings and Tijdeman's refined method—and shows that the latter achieves discrepancy near the optimal bound via a cut-and-project model with a polytopal window. Together, these results connect discrepancy theory, symbolic dynamics, and aperiodic order (model sets) to provide a rigorous, computable description of Tijdeman sequences in a dynamical systems framework, with explicit complexity bounds and spectral properties.

Abstract

In 1980, R. Tijdeman provided an on-line algorithm that generates sequences over a finite alphabet with minimal discrepancy, that is, such that the occurrence of each letter optimally tracks its frequency. In this article, we define discrete dynamical systems generating these sequences. The dynamical systems are defined as exchanges of polytopal pieces, yielding cut and project schemes, and they code tilings of the line whose sets of vertices form model sets. We prove that these sequences of low discrepancy are natural codings of toral translations with respect to polytopal atoms, and that they generate a minimal and uniquely ergodic subshift with purely discrete spectrum. Finally, we show that the factor complexity of these sequences is of polynomial growth order , where is the cardinality of the alphabet.
Paper Structure (16 sections, 14 theorems, 111 equations, 8 figures)

This paper contains 16 sections, 14 theorems, 111 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Notation and basic definitions
  3. Word combinatorics and symbolic dynamics
  4. Exchange of pieces, toral translations and natural codings
  5. Hypercubic fundamental domains
  6. Construction of natural partitions
  7. Two constructions of sequences with small discrepancy
  8. Strategy
  9. Hypercubic billiard sequences
  10. Tijdeman's construction
  11. Proof of Theorem \ref{['theo:main']}
  12. General strategy for estimating the factor complexity
  13. Upper bound for the factor complexity
  14. Lower bound for the factor complexity
  15. End of the proof of Theorem \ref{['theo:main']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\boldsymbol{\alpha} = (\alpha_1, \dots, \alpha_d)$ be a totally irrational frequency vector. Then there exist Tijdeman parameters generating a sequence $u$ with $\Delta_{\boldsymbol{\alpha}}(u) \leq 1{-}\tfrac{1}{2d-2}$, and such that $u$ is the bounded natural coding of $T_{\boldsymbol{\alpha}

Figures (8)

  • Figure 1: A fundamental domain of $\mathbb{R}^2 / \mathbb{Z}^2$ and its partition by finite unions of polygons such that the natural codings of the action of $T_{\boldsymbol{\alpha}}$ are Tijdeman sequences with $\boldsymbol{\alpha} \approx (0.5,0.45,0.05)$, $C = C' = D_3 = 3/4$.
  • Figure 2: A piece of a broken line associated with the sequence $12 11 212\cdots$.
  • Figure 3: Exchange of pieces, $d=2$.
  • Figure 4: The ($\iota$-representation of the) parallelepipeds $E_{\boldsymbol{\alpha},i}$ and their $\tilde{T}_{\boldsymbol{\alpha}}$-images for $d=3$, $\boldsymbol{\alpha} = (0.48,0.32,0.2)$.
  • Figure 5: Notation for hypercubic billiard sequences, with $\boldsymbol{\alpha} \approx (0.75,0.25)$, $\mathbf{x} \approx (0.2,0.6)$, thus $\mathbf{x}_0 \approx (-0.4,0.4)$. If $\mathbf{x}_n\in \{(-a,a): -\alpha_1< a<1-2\alpha_1 \}$, then $u_n=2$, and if $\mathbf{x}_n\in \{(-a,a): 1-2\alpha_1< a< 1-\alpha_1\}$, then $u_n=1$. Here $u_0u_1\cdots = 1211\dots$.
  • ...and 3 more figures

Theorems & Definitions (38)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Definition 2.2: Fundamental domains and natural partitions
  • Definition 2.3: Natural coding
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • Remark 2.7
  • Proposition 2.8
  • ...and 28 more