Dimension Spectrum of Continued fraction Expansions with Coefficients restricted to the Fibonacci Sequence

Arpit Dawar; Amit Priyadarshi

paper

Dimension Spectrum of Continued fraction Expansions with Coefficients restricted to the Fibonacci Sequence

arXiv:2606.26747

Abstract

In this paper, we analyze the structure of the dimension spectrum of continued fraction expansions with coefficients restricted to the generalized Fibonacci sequence. Let $F_{(a_{1}, a_{2})}$ denote the generalized Fibonacci sequence starting with the positive integers $a_{1} < a_{2}$ . We prove that the continued fractions whose digits lie in $F_{(a_{1}, a_{2})}$ have full dimension spectrum for every $(a_{1}, a_{2})$ such that $a_{1} \geq 2$ , or $a_{1} = 1$ and $a_{2} \geq 3$ . On the other hand, using the numerical tools developed by Falk and Nussbaum, we show that the dimension spectrum has a gap for continued fractions with digits restricted to each of the sets $F_{(1, 2)}$ and $F_{(2, 1)}$ , where $F_{(2, 1)}$ denotes the set of Lucas numbers. Moreover, for $F_{(1, 2)}$ and $F_{(2, 1)}$ , we prove that the dimension spectrum always contains a non-trivial interval.