kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2024-21-2-116-126kaz29-1259Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESГЕОМЕТРИЯ ФРАКТАЛОВ И МНОЖЕСТВА УРОВНЕЙ В ЦЕПНЫХ ДРОБЯХFRACTAL GEOMETRY AND LEVEL SETS INCONTINUED FRACTIONShttps://orcid.org/0000-0002-9658-9723КазинА.KazinA.

бакалавр

040900, г. Каскелен

Bachelor

040900, Kaskelen

aiken.kazin@sdu.edu.kzhttps://orcid.org/0000-0002-8352-2597КадыровШ.KadyrovSh.

PhD

г. Ташкент

PhD

Tashkent

shirali.kadyrov@oxusuniversity.uzУниверситет СДУКазахстанSDU UniversityKazakhstanУниверситет ОксусУзбекистанOxus UniversityUzbekistan202401072024212116126Copyright © Казин А., Кадыров Ш., 20242024Казин А., Кадыров Ш.Kazin A., Kadyrov S.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.kbtu.edu.kz/jour/article/view/1259

Цепные дроби предлагают уникальное представление действительных чисел в виде последовательности натуральных чисел. Основополагающая работа Гуда о цепных дробях положила начало дальнейшим исследованиям фрактальной геометрии и исключительных множеств. Эта статья расширяет выводы Гуда, сосредоточив внимание на множествах уровня, построенных путем ограничения частичных дробей нижними границами. Используя элементарные подходы, мы устанавливаем новые границы их хаусдорфовой размерности, предоставляя теоретические знания и практические методы оценки. Кроме того, мы предлагаем альтернативные доказательства и следствия, которые углубляют наше понимание взаимосвязи между цепными дробями и фрактальной геометрией. Цепные дроби представляют особый способ выражения действительных чисел в виде последовательности натуральных чисел, что позволяет лучше понять основную структуру этих чисел. Основываясь на фундаментальных исследованиях Гуда в области цепных дробей, эта статья углубляется в область фрактальной геометрии и исключительных множеств, исследуя интересные связи между этими математическими конструкциями. Наше внимание сосредоточено на исследовании хаусдорфовой размерности множеств уровня, образованных путем ограничения частичных дробей нижними границами. Используя элементарные методологии, мы представляем новые теоретические границы хаусдорфовой размерности этих множеств уровней, обогащая наше понимание их геометрических свойств. Сочетая теоретические достижения и практические методы, это исследование вносит вклад в математику, предоставляя как глубокие теоретические знания, так и практические приложения для понимания цепных дробей и их геометрических свойств.

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.

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The authors declare that there are no conflicts of interest present.