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Initial nonrepetitive complexity of regular episturmian words and their Diophantine exponents




TekijätPeltomäki Jarkko

KustantajaAcademic Press

Julkaisuvuosi2024

JournalEuropean Journal of Combinatorics

Artikkelin numero103942

Vuosikerta118

DOIhttps://doi.org/10.1016/j.ejc.2024.103942

Verkko-osoitehttps://doi.org/10.1016/j.ejc.2024.103942

Rinnakkaistallenteen osoitehttps://research.utu.fi/converis/portal/detail/Publication/387396181


Tiivistelmä

Regular episturmian words are episturmian words whose directive words have a regular and restricted form making them behave more like Sturmian words than general episturmian words. We present a method to evaluate the initial nonrepetitive complexity of regular episturmian words extending the work of Wojcik on Sturmian words. For this, we develop a theory of generalized Ostrowski numeration systems and show how to associate with each episturmian word a unique sequence of numbers written in this numeration system.

The description of the initial nonrepetitive complexity allows us to obtain novel results on the Diophantine exponents of regular episturmian words. We prove that the Diophantine exponent of a regular episturmian word is finite if and only if its directive word has bounded partial quotients. Moreover, we prove that the Diophantine exponent of a regular episturmian word is strictly greater than $2$ if the sequence of partial quotients is eventually at least $3$.

Given an infinite word $x$ over an integer alphabet, we may consider a real number $\xi_x$ having $x$ as a fractional part. The Diophantine exponent of $x$ is a lower bound for the irrationality exponent of $\xi_x$. Our results thus yield nontrivial lower bounds for the irrationality exponents of real numbers whose fractional parts are regular episturmian words. As a consequence, we identify a new uncountable class of transcendental numbers whose irrationality exponents are strictly greater than $2$. This class contains an uncountable subclass of Liouville numbers.


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Last updated on 2024-26-11 at 14:01