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

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 $a_1 \geq2$, or $a_1=1$ and $a_2\geq3$. 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.