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

TL;DR

This paper explores the dynamical systems underlying Tijdeman's minimal discrepancy sequences, revealing their structure as codings of toral translations with polynomial complexity growth.

Contribution

It introduces a dynamical systems framework for Tijdeman's sequences, linking them to polytopal exchanges, tilings, and model sets, and analyzes their spectral and complexity properties.

Findings

Sequences are natural codings of toral translations.

Generated systems are minimal and uniquely ergodic.

Sequence complexity grows polynomially as n^{d-1}."

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.