Sturmian numeration systems and decompositions to palindromes

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.

Submitted to European Journal of Combinatorics

Cited by in corpus (10)

• Palindromic Length and Reduction of Powers
• Palindromic length complexity and a generalization of Thue-Morse sequences
• The number of valid factorizations of Fibonacci prefixes
• Palindromic Length of Words with Many Periodic Palindromes
• Palindromic length sequence of the ruler sequence and of the period-doubling sequence
• Palindromic length of words and morphisms in class $\mathcal{P}$
• Quelques méthodes pour les mots sturmiens
• Prefix palindromic length of the Thue-Morse word
• A note on palindromic length of Sturmian sequences
• Palindromic length of infinite aperiodic words