Sturmian numeration systems and decompositions to palindromes

TL;DR

This paper introduces an extended Ostrowski numeration system related to Sturmian words, enabling better detection of palindromes and proving a conjecture that non-periodic Sturmian words have prefixes resistant to decomposition into few palindromes.

Contribution

It extends classical Ostrowski numeration systems to better reflect Sturmian word structures and proves a conjecture about the palindromic complexity of non-periodic Sturmian words.

Findings

Extended numeration system captures palindrome occurrences in Sturmian words

Proved that non-periodic Sturmian words have prefixes with high palindromic complexity

Established a new link between numeration systems and combinatorics on words

Abstract

We extend the classical Ostrowski numeration systems, closely related to Sturmian words, by allowing a wider range of coefficients, so that possible representations of a number $n$ better reflect the structure of the associated Sturmian word. In particular, this extended numeration system helps to catch occurrences of palindromes in a characteristic Sturmian word and thus to prove for Sturmian words the following conjecture stated in 2013 by Puzynina, Zamboni and the author: If a word is not periodic, then for every $Q>0$ it has a prefix which cannot be decomposed to a concatenation of at most $Q$ palindromes.