On the dimension spectrum of infinite subsystems of continued fractions

Abstract

In this paper we study the dimension spectrum of continued fractions with coefficients restricted to infinite subsets of natural numbers. We prove that if E E is any arithmetic progression, the set of primes, or the set of squares { n 2 } n ∈ N \{n^2\}_{n \in \mathbb {N}} , then the continued fractions whose digits lie in E E have full dimension spectrum, which we denote by D S ( C F E ) DS(\mathcal {CF}_E) . Moreover we prove that if E E is an infinite set of consecutive powers, then the dimension spectrum D S ( C F E ) DS(\mathcal {CF}_E) always contains a non-trivial interval. We also show that there exists some E ⊂ N E \subset \mathbb {N} and two non-trivial intervals I 1 , I 2 I_1, I_2 , such that D S ( C F E ) ∩ I 1 = I 1 DS(\mathcal {CF}_E) \cap I_1=I_1 and D S ( C F E ) ∩ I 2 DS(\mathcal {CF}_E) \cap I_2 is a Cantor set. On the way we employ the computational approach of Falk and Nussbaum in order to obtain rigorous effective estimates for the Hausdorff dimension of continued fractions whose entries are restricted to infinite sets.