On balance properties of hypercubic billiard words

TL;DR

This paper investigates the balance properties of hypercubic billiard words, revealing that while some factors are balanced, others exhibit unbounded imbalance, using topological and arithmetic methods.

Contribution

It introduces the first analysis of balance properties in hypercubic billiard words, showing the existence of both balanced and unbalanced factors.

Findings

Number of letter occurrences in factors depends only on length, up to a constant

Existence of unbalanced factors with unbounded letter count differences

Use of topological and arithmetic methods for proofs

Abstract

This paper studies balance properties for billiard words. Billiard words generalize Sturmian words by coding trajectories in hypercubic billiards. In the setting of aperiodic order, they also provide the simplest examples of quasicrystals, as tilings of the line obtained via cut and project sets with a cubical canonical window. By construction, the number of occurrences of each letter in a factor (i.e., a string of consecutive letters) of a hypercubic billiard word only depends on the length of the factor, up to an additive constant. In other words, the difference of the number of occurrences of each letter in factors of the same length is bounded. In contrast with the behaviour of letters, we prove the existence of words that are not balanced in billiard words: the difference of the number of occurrences of such unbalanced factors in longer factors of the same length is unbounded. The proof relies both on topological methods inspired by tiling cohomology and on arithmetic results on bounded remainder sets for toral translations.