On matching and periodicity for (N,α)-expansions
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In: The Ramanujan Journal , Vol. 64.2024, No. 4, 08.2024, p. 1479-1496.
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TY - JOUR
T1 - On matching and periodicity for (N,α)-expansions
AU - Kraaikamp, Cor
AU - Langeveld, Niels
N1 - Publisher Copyright: © The Author(s) 2024.
PY - 2024/8
Y1 - 2024/8
N2 - Recently a new class of continued fraction algorithms, the (N,α)-expansions, was introduced in Kraaikamp and Langeveld (J Math Anal Appl 454(1):106–126, 2017) for each N∈N, N≥2 and α∈(0,N-1]. Each of these continued fraction algorithms has only finitely many possible digits. These (N,α)-expansions ‘behave’ very different from many other (classical) continued fraction algorithms; see also Chen and Kraaikamp (Matching of orbits of certain n-expansions with a finite set of digits (2022). To appear in Tohoku Math. J arXiv:2209.08882), de Jonge and Kraaikamp (Integers 23:17, 2023), de Jonge et al. (Monatsh Math 198(1):79–119, 2022), Nakada (Tokyo J Math 4(2):399–426, 1981) for examples and results. In this paper we will show that when all digits in the digit set are co-prime with N, which occurs in specified intervals of the parameter space, something extraordinary happens. Rational numbers and certain quadratic irrationals will not have a periodic expansion. Furthermore, there are no matching intervals in these regions. This contrasts sharply with the regular continued fraction and more classical parameterised continued fraction algorithms, for which often matching is shown to hold for almost every parameter. On the other hand, for α small enough, all rationals have an eventually periodic expansion with period 1. This happens for all α when N=2. We also find infinitely many matching intervals for N=2, as well as rationals that are not contained in any matching interval.
AB - Recently a new class of continued fraction algorithms, the (N,α)-expansions, was introduced in Kraaikamp and Langeveld (J Math Anal Appl 454(1):106–126, 2017) for each N∈N, N≥2 and α∈(0,N-1]. Each of these continued fraction algorithms has only finitely many possible digits. These (N,α)-expansions ‘behave’ very different from many other (classical) continued fraction algorithms; see also Chen and Kraaikamp (Matching of orbits of certain n-expansions with a finite set of digits (2022). To appear in Tohoku Math. J arXiv:2209.08882), de Jonge and Kraaikamp (Integers 23:17, 2023), de Jonge et al. (Monatsh Math 198(1):79–119, 2022), Nakada (Tokyo J Math 4(2):399–426, 1981) for examples and results. In this paper we will show that when all digits in the digit set are co-prime with N, which occurs in specified intervals of the parameter space, something extraordinary happens. Rational numbers and certain quadratic irrationals will not have a periodic expansion. Furthermore, there are no matching intervals in these regions. This contrasts sharply with the regular continued fraction and more classical parameterised continued fraction algorithms, for which often matching is shown to hold for almost every parameter. On the other hand, for α small enough, all rationals have an eventually periodic expansion with period 1. This happens for all α when N=2. We also find infinitely many matching intervals for N=2, as well as rationals that are not contained in any matching interval.
KW - N-continued fractions
KW - quadratic irrationals
KW - matching
KW - periodicity
KW - Matching
KW - 37A45
KW - Quadratic irrationals
KW - Periodicity
KW - 37E15
KW - 37E05
KW - (N,α)-continued fractions
UR - http://www.scopus.com/inward/record.url?scp=85194541106&partnerID=8YFLogxK
U2 - 10.1007/s11139-024-00878-7
DO - 10.1007/s11139-024-00878-7
M3 - Article
VL - 64.2024
SP - 1479
EP - 1496
JO - The Ramanujan Journal
JF - The Ramanujan Journal
SN - 1572-9303
IS - 4
ER -