Gaps in intervals of N-expansions

Journal article (2023)

Authors

C.J. de Jonge Applied Probability - EEMCS , Korteweg-de Vries Institute for Mathematics
Cor Kraaikamp Universiteit van Amsterdam
Research Group
Applied Probability (EEMCS) (TU Delft)
Copyright: © 2023 C.J. de Jonge, Cor Kraaikamp
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Published Date

2023

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Applied Probability

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.

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