<p>Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$x \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be an irrational number with continued fraction expansion <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$[a_1(x),a_2(x), \cdots ,a_n(x),\cdots ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mo>⋯</mo> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\psi : \mathbb {R}_{>0}\rightarrow \mathbb {R}_{>0}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo>:</mo> <msub> <mi mathvariant="double-struck">R</mi> <mrow> <mo>></mo> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi mathvariant="double-struck">R</mi> <mrow> <mo>></mo> <mn>0</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> be a non-increasing function. In this paper, we mainly study the intersection of the set of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\psi -$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>well approximated numbers and the level sets of the convergence exponent in continued fraction expansions. It is proved that the Hausdorff dimension of their intersections equals the product of their Hausdorff dimensions.</p>

On the convergence exponent of $\psi -$well approximated numbers

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发表日期：

2026年1月16日

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Zhenliang Zhang

摘要

Let $x \in (0,1)$ $x \in (0, 1)$ be an irrational number with continued fraction expansion $[a_1(x),a_2(x), \cdots ,a_n(x),\cdots ]$ $[a_{1} (x), a_{2} (x), \dots, a_{n} (x), \dots]$ and $\psi : \mathbb {R}_{>0}\rightarrow \mathbb {R}_{>0}$ $ψ : R_{> 0} \to R_{> 0}$ be a non-increasing function. In this paper, we mainly study the intersection of the set of $\psi -$ $ψ -$ well approximated numbers and the level sets of the convergence exponent in continued fraction expansions. It is proved that the Hausdorff dimension of their intersections equals the product of their Hausdorff dimensions.

On the convergence exponent of \(\psi -\)well approximated numbers

摘要