Sturmian numeration systems and decompositions to palindromes

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 words 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.