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arXiv · 2606.26747

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

Arpit Dawar·Amit Priyadarshi
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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.

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Arpit Dawar, Amit Priyadarshi. 2026-06-25. Dimension Spectrum of Continued fraction Expansions with Coefficients restricted to the Fibonacci Sequence. https://arxiv.org/abs/2606.26747

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