For any x in (0, 1], let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$[d_1(x),d_2(x),\cdots ,d_n(x),\cdots ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>d</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>d</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> be its Lüroth expansion. In this paper, our objective is to investigate the multifractal spectrum of the intersection of the level sets of the convergence exponent in Lüroth expansion and set of points with prescribed relative growth rates of digits in the Lüroth expansion compared to convergence speed or denominators of its convergents. More precisely, for any <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha \ge 0,\beta \ge 0, z\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mi>β</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mi>z</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Delta (\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the level sets of the convergence exponent in Lüroth expansion and B(z), <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$U(\beta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the sets of points with prescribed relative growth rates of digits in the Lüroth expansion compared to convergence speed or denominators of its convergents, respectively. Then the Hausdorff dimensions of the intersection sets admit a dichotomy. Moreover, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\dim _H\Delta (\alpha )\cap B(z) = \dim _H\Delta (\alpha )\times \dim _HB(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>dim</mo> <mi>H</mi> </msub> <mi mathvariant="normal">Δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo>dim</mo> <mi>H</mi> </msub> <mi mathvariant="normal">Δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msub> <mo>dim</mo> <mi>H</mi> </msub> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, but <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\dim _H\Delta (\alpha )\cap U(\beta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>dim</mo> <mi>H</mi> </msub> <mi mathvariant="normal">Δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <mi>U</mi> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> exhibits different phenomenon.

On Irrationals with Prescribed Relative Growth Rates and Convergence Exponents for Lüroth Expansions

期刊论文
发表日期：

2026年1月20日

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Zhenliang Zhang,
Weidong Li

摘要

For any x in (0, 1], let $[d_1(x),d_2(x),\cdots ,d_n(x),\cdots ]$ $[d_{1} (x), d_{2} (x), \dots, d_{n} (x), \dots]$ be its Lüroth expansion. In this paper, our objective is to investigate the multifractal spectrum of the intersection of the level sets of the convergence exponent in Lüroth expansion and set of points with prescribed relative growth rates of digits in the Lüroth expansion compared to convergence speed or denominators of its convergents. More precisely, for any $\alpha \ge 0,\beta \ge 0, z\ge 0$ $α \geq 0, β \geq 0, z \geq 0$ , let $\Delta (\alpha )$ $Δ (α)$ be the level sets of the convergence exponent in Lüroth expansion and B(z), $U(\beta )$ $U (β)$ be the sets of points with prescribed relative growth rates of digits in the Lüroth expansion compared to convergence speed or denominators of its convergents, respectively. Then the Hausdorff dimensions of the intersection sets admit a dichotomy. Moreover, $\dim _H\Delta (\alpha )\cap B(z) = \dim _H\Delta (\alpha )\times \dim _HB(z)$ ${dim}_{H} Δ (α) \cap B (z) = {dim}_{H} Δ (α) \times {dim}_{H} B (z)$ , but $\dim _H\Delta (\alpha )\cap U(\beta )$ ${dim}_{H} Δ (α) \cap U (β)$ exhibits different phenomenon.