On the structure of the dimension spectrum for continued fraction expansions

Painos Chitanga; Bas Lemmens; Roger Nussbaum

doi:10.1016/j.indag.2026.03.008

paper

On the structure of the dimension spectrum for continued fraction expansions

arXiv:2504.20878 · doi:10.1016/j.indag.2026.03.008

Abstract

We analyse the dimension spectrum of continued fractions expansions with coefficients restricted to infinite subsets of $N$ . We prove that the set of powers $P_{q} = {q^{n} : n \in N}$ has full dimension spectrum for each integer $q \geq 2$ , answering a question by Chousionis, Leykekhman and Urbański. On the other hand, we show that the dimension spectrum for $P_{q}^{*} = {q^{n} : n \in N} \cup {1}$ has many gaps and regions where it is nowhere dense. We also investigate the case where $A$ is generated by a monomial, $M_{q} = {n^{q} : n \in N}$ . For $M_{q}$ we prove that the dimension spectrum is full for $q \in {1, 2, 3, 4, 5}$ , and it has a gap for each $q \geq 6$ . Furthermore we show for $q \in {6, 7, 8}$ that the dimension spectrum of $M_{q}$ is the disjoint union of two nontrivial closed intervals, and it is the disjoint union of three nontrivial closed intervals for $q \in {9, 10}$ . For $q \geq 11$ we show that the dimension spectrum of $M_{q}$ consists of finitely many disjoint nontrivial closed intervals. The results concerning $M_{q}$ extend existing results for $q = 1$ and $q = 2$ . In our analysis we employ Perron-Frobenius (transfer) operators, and numerical tools developed by Falk and Nussbaum that give rigorous estimates for the Hausdorff dimension for continued fractions expansions.

27 pages, added some references and made minor corrections to previous version