On the structure of the dimension spectrum for continued fraction expansions

TL;DR

This paper analyzes the dimension spectrum $\text{DS}(A)$ for continued fraction sets $J_A$ generated by infinite subsets $A\subset \mathbb{N}$. Using Perron–Frobenius transfer operators $L_{s,F}$ and numerically rigorous Falk–Nussbaum bounds, it proves that $A=\{q^n\}$ has full dimension spectrum, while $A=\{q^n\colon n\ge1\}\cup\{1\}$ (denoted $P_q^*$) exhibits gaps and nowhere-dense regions. It further determines the dimension spectrum for monomial-generated sets $M_q=\{n^q\}$: full for $q\in\{1,2,3,4,5\}$, with a sequence of interval decompositions appearing for larger $q$ and finitely many disjoint intervals for all $q\ge 11$. The analysis hinges on a robust combination of strict break-point arguments and transfer-operator spectral bounds, enabling both qualitative gap-structure statements and quantitative interval descriptions. These results extend prior special cases ($q=1,2$) and contribute a general criterion for when the dimension spectrum decomposes into finitely many closed intervals, with implications for Texan-type conjectures and fractal dimensions in continued fraction dynamics.

Abstract

We analyse the dimension spectrum of continued fractions expansions with coefficients restricted to infinite subsets of $ \mathbb{N}$. We prove that the set of powers $P_q=\{q^n\colon n\in \mathbb{N}\}$ 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 $P^*_q=\{q^n\colon n\in \mathbb{N}\}\cup\{1\}$ has many gaps and regions where it is nowhere dense. We also investigate the case where $A$ is generated by a monomial, $M_q=\{n^q\colon n\in\mathbb{N}\}$. 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.

Theorems & Definitions (55)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof