A new class of α -Farey maps and an application to normal numbers

Karma Dajani; Cornelis Kraaikamp; Hitoshi Nakada; Rie Natsui

doi:10.1017/prm.2025.10068

A new class of α -Farey maps and an application to normal numbers

Karma Dajani, Cornelis Kraaikamp^*, Hitoshi Nakada, Rie Natsui

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We define two types of the α-Farey maps Fα and for, which were previously defined only for by Natsui (2004). Then, for each, we construct the natural extension maps on the plane and show that the natural extension of is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associated with α-continued fractions does not vary by the choice of α,. This extends the result by Kraaikamp and Nakada (2000).

Original language	English
Number of pages	37
Journal	Proceedings of the Royal Society of Edinburgh Section A: Mathematics
DOIs	https://doi.org/10.1017/prm.2025.10068
Publication status	E-pub ahead of print - 15 Sept 2025

Bibliographical note

Publisher Copyright:
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

Funding

This research was partially supported by JSPS grants 20K03642 and 24K06785. We thank the anonymous referee whose comments greatly improved the exposition of this paper.

Funders	Funder number
JSPS	20K03642, 24K06785

Keywords

Farey map
natural extension
normal numbers
α-continued fraction expansions

Access to Document

10.1017/prm.2025.10068Licence: CC BY

Cite this

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abstract = "We define two types of the α-Farey maps Fα and for, which were previously defined only for by Natsui (2004). Then, for each, we construct the natural extension maps on the plane and show that the natural extension of is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associated with α-continued fractions does not vary by the choice of α,. This extends the result by Kraaikamp and Nakada (2000).",

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AU - Dajani, Karma

AU - Kraaikamp, Cornelis

AU - Nakada, Hitoshi

AU - Natsui, Rie

PY - 2025/9/15

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N2 - We define two types of the α-Farey maps Fα and for, which were previously defined only for by Natsui (2004). Then, for each, we construct the natural extension maps on the plane and show that the natural extension of is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associated with α-continued fractions does not vary by the choice of α,. This extends the result by Kraaikamp and Nakada (2000).

AB - We define two types of the α-Farey maps Fα and for, which were previously defined only for by Natsui (2004). Then, for each, we construct the natural extension maps on the plane and show that the natural extension of is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associated with α-continued fractions does not vary by the choice of α,. This extends the result by Kraaikamp and Nakada (2000).

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KW - normal numbers

KW - α-continued fraction expansions

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JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

ER -

A new class of α -Farey maps and an application to normal numbers

Abstract

Bibliographical note

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Keywords

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A new class of α-Farey maps and an application to normal numbers

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