Fractal geometry of continued fractions with large coefficients and dimension drop problems

Abstract: In 1928, Jarn\'{\i}k \cite{Jar} obtained that the set of continued fractions with bounded coefficients has Hausdorff dimension one. Good \cite{Goo} observed a dimension drop phenomenon by proving that the Hausdorff dimension of the set of continued fractions whose coefficients tend to infinity is one-half. For the set of continued fractions whose coefficients tend to infinity rapidly, Luczak \cite{Luc} and Feng et al. \cite{FWLT} showed that its Hausdorff dimension decreases even further. Recently, Liao and Rams \cite{LR16} also observed an analogous dimension drop phenomenon when they studied the subexponential growth rate of the sum of coefficients. In this paper, we consolidate and considerably extend the studies of the abovementioned problem into a general dimension drop problem on the distribution of continued fractions with large coefficients. As applications, we use a different approach to reprove a result of Wang and Wu on the dimensions of the Borel-Bernstein sets \cite{WW}, fulfil the dimension gap proposed by Liao and Rams \cite{LR16}, and establish several new results concerning the dimension theory of liminf and limsup sets related to the maximum of coefficients.

• The paper formulates a thermodynamic formalism to study Hausdorff dimensions in continued fractions with large coefficients.
• It establishes quantitative conditions where dimensions drop significantly, including cases from full dimension to zero.
• The study extends earlier works and applies novel techniques to address longstanding dimension drop problems in fractal geometry.

Fractal Geometry of Continued Fractions with Large Coefficients and Dimension Drop Problems

The study explores critical aspects of the fractal geometry of continued fractions, particularly focusing on the dimension drop phenomenon associated with the asymptotic behavior of continued fractions when their coefficients are large. This paper is authored by Lulu Fang, Carlos Gustavo Moreira, and Yiwei Zhang, building upon foundational principles laid out by Jarník, Good, and Liao and Rams regarding continued fractions and their Hausdorff dimensions.

Continued fractions are a fundamental aspect of number theory and exhibit connections to various mathematical domains, including dynamical systems and fractal geometry. The paper significantly extends previous studies by presenting a general framework to examine the dimension drop problem in continued fractions with large coefficients. It explores the set $E(\{n_k\},\{s_k\}, \{t_k\})$, characterized by a sequence of continued fraction coefficients where certain conditions on the indices $n_k$ and the growth sequences $s_k$, $t_k$ apply.

Main Contributions

• Formulation in Thermodynamic Formalism: The authors employ thermodynamic formalism to describe the Hausdorff dimension of continued fraction sets under varying conditions. Specifically, they analyze the dimension of the set $E(\{n_k\},\{s_k\}, \{t_k\})$ using Diophantine pressure equations and provide a comprehensive description based on the values of the key parameters $\alpha$ and $\beta$. This approach provides a mathematical framework to understand dimension variability with large coefficients.
• Novel Results on Hausdorff Dimensions: The findings reveal that for different asymptotic properties of coefficient sequences, the dimension of the set can vary from full dimension to significant drop-offs, including cases yielding dimension zero. Notably, they identify and quantify conditions where dimension drops from one to one-half, and further, cases where it drops to zero, providing detailed quantitative insights on this phenomenon.
• Applications to Sum and Maximum of Coefficients: The research also addresses diverse applications related to sums and maximums of coefficients in continued fraction expansions. The analysis revisits and re-proves the dimension of Borel-Bernstein sets, resolves dimension gaps first observed by Liao and Rams, and presents new results concerning liminf and limsup sets related to coefficient maxima.

Implications and Future Directions

The implications of this work extend beyond theoretical curiosity, with potential applications to broader areas of number theory and dynamics, including Diophantine approximation and ergodic theory. The findings could influence computational approaches in assessing number systems' detailed structures, impacting algorithms dependent on the arithmetic properties of real numbers.

The paper opens several avenues for future exploration:

• Further refinements in the estimation of Hausdorff dimensions for more complex sequences of partial quotients, enhancing the precision in predicting dimension behavior under varied fraction expansions.
• The implications of these findings in dynamical systems, particularly in understanding measure-theoretic properties of transformation spaces underlying continued fractions.
• Expanding on numerical simulations and computational experiments, validating these rigorous mathematical results in practical scenarios.

The authors' approach and results contribute significantly to continued fractions' metrical theory, offering advanced insights into fractal properties that could spark new investigations in mathematical analysis and its applications.