On the dimension spectrum of infinite subsystems of continued fractions

TL;DR

This paper investigates the Hausdorff dimension spectrum of continued fractions with digits in various infinite subsets of natural numbers, revealing full spectra for certain sets and complex structures for others.

Contribution

It establishes full dimension spectrum results for specific infinite digit sets and demonstrates the existence of non-trivial intervals and Cantor set structures within the spectrum.

Findings

Full spectrum for arithmetic progressions, primes, and squares.

Presence of non-trivial intervals in the spectrum for powers.

Existence of Cantor set structures in the spectrum.

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 $E$ is any arithmetic progression, the set of primes, or the set of squares $\{n^2\}_{n \in \mathbb{N}}$, then the continued fractions whose digits lie in $E$ have full dimension spectrum, which we denote by $DS(\mathcal{CF}_E)$. Moreover we prove that if $E$ is an infinite set of consecutive powers then the dimension spectrum $DS(\mathcal{CF}_E)$ always contains a non trivial interval. We also show that there exists some $E \subset \mathbb{N}$ and two non-trivial intervals $I_1, I_2$, such that $DS(\mathcal{CF}_E) \cap I_1=I_1$ and $DS(\mathcal{CF}_E) \cap I_2$ 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.