On matching and periodicity for $$(N,\alpha )$$-expansions

Abstract

AbstractRecently a new class of continued fraction algorithms, the $$(N,\alpha $$ ( N , α )-expansions, was introduced in Kraaikamp and Langeveld (J Math Anal Appl 454(1):106–126, 2017) for each $$N\in \mathbb {N}$$ N ∈ N , $$N\ge 2$$ N ≥ 2 and $$\alpha \in (0,\sqrt{N}-1]$$ α ∈ ( 0 , N - 1 ] . Each of these continued fraction algorithms has only finitely many possible digits. These $$(N,\alpha )$$ ( 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 $$\alpha $$ α small enough, all rationals have an eventually periodic expansion with period 1. This happens for all $$\alpha $$ α when $$N=2$$ N = 2 . We also find infinitely many matching intervals for $$N=2$$ N = 2 , as well as rationals that are not contained in any matching interval.