FRACTAL GEOMETRY AND LEVEL SETS INCONTINUED FRACTIONS

Abstract

Continued fractions offer a unique representation of real numbers as a sequence of natural numbers. Good's seminal work on continued fractions laid further research into fractal geometry and exceptional sets. This paper extends Good's findings by focusing on level sets constructed by restricting the partial quotients with lower bounds. Using elementary approaches, we establish new bounds on their Hausdorff dimension, providing theoretical insights and practical estimation methods. Additionally, we offer alternative proofs and corollaries that deepen our understanding of the relationship between continued fractions and fractal geometry. Continued fractions provide a distinctive means of expressing real numbers as a sequence of natural numbers, offering insights into the underlying structure of these numbers. Building upon Good's foundational research in continued fractions, this paper delves into the domain of fractal geometry and exceptional sets, exploring the interesting connections between these mathematical constructs. Our focus lies on investigating the Hausdorff dimension of level sets formed by constraining the partial quotients with lower bounds. Employing elementary methodologies, we present fresh theoretical bounds on Hausdorff dimension of these level sets, enriching our understanding of their geometric properties. Through combining theoretical advancements and practical techniques, this research contributes to mathematics, providing both deep theoretical insights and practical applications in understanding continued fractions and their geometric properties.