Browsing by Author "Duisen S."

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    Fractal dimension of exceptional sets in semi-regular continued fraction
    (SDU University, 2025) Duisen S.
    This thesis investigates the interplay between Diophantine approximation, continued fraction representations, and fractal geometry. We begin by exploring the classical notion of badly approximable numbers-real numbers whose continued fraction expansions have bounded partial quotients. These numbers, while forming a set of zero Lebesgue measure, exhibit full Hausdorff dimension, highlighting their rich geometric structure. Building on this foundation, we introduce and analyze a generalization known as semi-regular continued fractions, wherein a fixed sequence of signs modifies the classical expansion. For such expansions, we define the class of σ-badly approximable numbers and study their distribution and fractal properties. We demonstrate that these generalized expansions preserve many of the geometric complexities of their classical counterparts, while offering new degrees of arithmetic freedom. In the second part of the thesis, we shift our focus to Lehner expansions of real numbers and examine how the statistical behavior of the associated digit sequence (bn) influences the fractal geometry of the corresponding number sets. Specifically, we investigate the impact of the average value of bn on the box dimension-a quantitative measure of geometric complexity. Employing the box-counting method, we perform numerical experiments to estimate the box dimension and uncover how variations in the digit sequence relate to the irregularity and structure of the expansion. By synthesizing the analytical and numerical approaches, this thesis provides a comprehensive view of how modifications to continued fraction representations influence the fractal characteristics of real number sets, contributing to the broader understanding of number-theoretic and geometric interrelations.
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    Fractal Dimension of Exceptional Sets in Semi-Regular Continued Fractions
    (faculty of engineering and natural sciences, 2025) Kadyrov Sh.; Kazin A.; Duisen S.
    This paper examines how the average value of the sequence bn in the Lehner expansion of a real number x influences its box dimension. Our primary objective is to analyze how variations in the average of bn impact the box dimension, which serves as a measure of the complexity of the sequence. Using the box-counting method, we numerically estimate the box dimension and explore its relationship with the fractal nature of Lehner expansions.