P. Erdős et al. proved in 1990 that every nontrivial number has a continuum of expansions on two-letter alphabets in every base smaller than the Golden ratio, and that this property fails for the Golden ratio base. It was shown in a recent paper of Baiocchi et al. that if we replace the powers of the Golden ratio by the closely related Fibonacci sequence, then the resulting integer base expansions still have the continuum expansion property. The proof heavily relied on the special properties of the Golden ratio. The difficulty came from the fact that the new expansions do not have any more the ergodic structure of non-integer base expansions. In this paper we introduce a new “quasi-ergodic” approach that allows us to handle many more general cases. We apply this approach to Baker's generalized Golden ratios.

A quasi-ergodic approach to non-integer base expansions / Komornik, V.; Loreti, P.; Pedicini, M.. - In: JOURNAL OF NUMBER THEORY. - ISSN 0022-314X. - 254:(2024), pp. 146-168. [10.1016/j.jnt.2023.07.009]

A quasi-ergodic approach to non-integer base expansions

Loreti P.;
2024

Abstract

P. Erdős et al. proved in 1990 that every nontrivial number has a continuum of expansions on two-letter alphabets in every base smaller than the Golden ratio, and that this property fails for the Golden ratio base. It was shown in a recent paper of Baiocchi et al. that if we replace the powers of the Golden ratio by the closely related Fibonacci sequence, then the resulting integer base expansions still have the continuum expansion property. The proof heavily relied on the special properties of the Golden ratio. The difficulty came from the fact that the new expansions do not have any more the ergodic structure of non-integer base expansions. In this paper we introduce a new “quasi-ergodic” approach that allows us to handle many more general cases. We apply this approach to Baker's generalized Golden ratios.
2024
Beta-expansions; Lucas sequences
01 Pubblicazione su rivista::01a Articolo in rivista
A quasi-ergodic approach to non-integer base expansions / Komornik, V.; Loreti, P.; Pedicini, M.. - In: JOURNAL OF NUMBER THEORY. - ISSN 0022-314X. - 254:(2024), pp. 146-168. [10.1016/j.jnt.2023.07.009]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1704957
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