Bracket words: a generalisation of Sturmian words arising from generalised polynomials

TL;DR

This paper explores the combinatorial properties of bracket words, a class of sequences derived from generalised polynomials, establishing a polynomial bound on their subword complexity and connecting them to Sturmian words.

Contribution

It introduces bracket words as a new class of sequences from generalised polynomials and proves a polynomial bound on their subword complexity, extending understanding of their combinatorial structure.

Findings

Bracket words can be finitely valued but non-periodic.

Subword complexity of bracket words is polynomially bounded.

Connections between generalised polynomials and Sturmian words are established.

Abstract

Generalised polynomials are maps constructed by applying the floor function, addition, and multiplication to polynomials. Despite superficial similarity, generalised polynomials exhibit many phenomena which are impossible for polynomials. In particular, there exist generalised polynomial sequences which take only finitely many values without being periodic; examples of such sequences include the Sturmian words, as well as more complicated sequences like $[ 2\{ \pi n^2 + \sqrt{2}n[\sqrt{3}n] \}]$. The purpose of this paper is to investigate letter-to-letter codings of finitely-valued generalised polynomial sequences, which we call \emph{bracket words}, from the point of view of combinatorics on words. We survey existing results on generalised polynomials and their corollaries in terms of bracket words, and also prove several new results. Our main contribution is a polynomial bound on the subword complexity of bracket words.