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<p>This paper explores Diophantine approximation and its connection to continued fractions. A number is called “badly approximable” if its regular continued fraction expansion has bounded partial quotients. These numbers have interesting geometric properties, such as full Hausdorff dimension but zero Lebesgue measure. We extend this idea to semi-regular continued fractions, a variation that uses a sequence of signs to modify the representation. We define <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-badly approximable numbers as those with bounded partial quotients in a semi-regular continued fractions for a fixed sequence of signs <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>. This leads us to investigate the fractal properties of such numbers.</p>

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A note on semi-regular continued fractions with bounded partial quotients

  • 期刊论文

  • 发表日期：

    2026年1月27日

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  • S. Kadyrov,
  • A. Kazin,
  • S. Duisen

摘要

This paper explores Diophantine approximation and its connection to continued fractions. A number is called “badly approximable” if its regular continued fraction expansion has bounded partial quotients. These numbers have interesting geometric properties, such as full Hausdorff dimension but zero Lebesgue measure. We extend this idea to semi-regular continued fractions, a variation that uses a sequence of signs to modify the representation. We define \(\sigma \) σ -badly approximable numbers as those with bounded partial quotients in a semi-regular continued fractions for a fixed sequence of signs \(\sigma \) σ . This leads us to investigate the fractal properties of such numbers.