Gaps in intervals of N-expansions

Jaap de Jonge; Cor Kraaikamp

Gaps in intervals of N-expansions

Jaap de Jonge
, Cor Kraaikamp

Applied Probability

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

3 Citations (Scopus)

22 Downloads (Pure)

Abstract

For N ∈ N_≥2 and α ∈ R such that 0 < α ≤ N − 1, the continued fraction map T_α: [α, α+1] → [α, α+1) is defined as T_α (x):= N/x−d(x), where d: [α, α+1] → N is defined by d(x):= ⌊N/x − α⌋. A maximal open interval (a, b) ⊂ I_α is called a gap of I_α if for almost every x ∈ I_α there is an n₀ (x) ∈ N such that x_n /∈ (a, b) for all n ≥ n₀ . In this paper, all conditions are given in which I_α is gapless. For α =^√N − 1 it is shown that the number of gaps is a finite, monotonically non-decreasing and unbounded function of N.

Original language	English
Article number	A42
Number of pages	17
Journal	Integers
Volume	23
Publication status	Published - 2023

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Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care
Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

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title = "Gaps in intervals of N-expansions",

abstract = "For N ∈ N≥2 and α ∈ R such that 0 < α ≤ N − 1, the continued fraction map Tα: [α, α+1] → [α, α+1) is defined as Tα (x):= N/x−d(x), where d: [α, α+1] → N is defined by d(x):= ⌊N/x − α⌋. A maximal open interval (a, b) ⊂ Iα is called a gap of Iα if for almost every x ∈ Iα there is an n0 (x) ∈ N such that xn /∈ (a, b) for all n ≥ n0 . In this paper, all conditions are given in which Iα is gapless. For α =√N − 1 it is shown that the number of gaps is a finite, monotonically non-decreasing and unbounded function of N.",

author = "\{de Jonge\}, Jaap and Cor Kraaikamp",

note = "Green Open Access added to TU Delft Institutional Repository {\textquoteleft}You share, we take care!{\textquoteright} – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public. ",

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language = "English",

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journal = "Integers",

issn = "1867-0652",

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T1 - Gaps in intervals of N-expansions

AU - de Jonge, Jaap

AU - Kraaikamp, Cor

N1 - Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

PY - 2023

Y1 - 2023

N2 - For N ∈ N≥2 and α ∈ R such that 0 < α ≤ N − 1, the continued fraction map Tα: [α, α+1] → [α, α+1) is defined as Tα (x):= N/x−d(x), where d: [α, α+1] → N is defined by d(x):= ⌊N/x − α⌋. A maximal open interval (a, b) ⊂ Iα is called a gap of Iα if for almost every x ∈ Iα there is an n0 (x) ∈ N such that xn /∈ (a, b) for all n ≥ n0 . In this paper, all conditions are given in which Iα is gapless. For α =√N − 1 it is shown that the number of gaps is a finite, monotonically non-decreasing and unbounded function of N.

AB - For N ∈ N≥2 and α ∈ R such that 0 < α ≤ N − 1, the continued fraction map Tα: [α, α+1] → [α, α+1) is defined as Tα (x):= N/x−d(x), where d: [α, α+1] → N is defined by d(x):= ⌊N/x − α⌋. A maximal open interval (a, b) ⊂ Iα is called a gap of Iα if for almost every x ∈ Iα there is an n0 (x) ∈ N such that xn /∈ (a, b) for all n ≥ n0 . In this paper, all conditions are given in which Iα is gapless. For α =√N − 1 it is shown that the number of gaps is a finite, monotonically non-decreasing and unbounded function of N.

UR - https://www.scopus.com/pages/publications/85163355209

M3 - Article

AN - SCOPUS:85163355209

SN - 1867-0652

VL - 23

JO - Integers

JF - Integers

M1 - A42

ER -

Gaps in intervals of N-expansions

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