Let [d1(x), d2(x), ⋯] be the Lüroth expansion of any real number x ∈ (0, 1]. Fix c ≥ 1 and let Φ: ℕ → (1, ∞) be a positive function. We calculate the Lebesgue measure dichotomy statement (a zero-one law) and the Hausdorff dimension of the following set <Equation ID="Equ1"> <EquationSource Format="TEX">${\Lambda _c}(\Phi ): = \left\{ {x \in \left( {0,1} \right]:{d_n}(x){d_{2n}}(x) \cdots {d_{cn}}(x) \geqslant \Phi (n)\;{\rm{for\; infinitely\;many}}\;n \in \mathbb{N}} \right\}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Λ</mi> <mi>c</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>d</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>n</mi> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⩾</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">y</mi> <mspace width="thickmathspace" /> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">y</mi> </mrow> </mrow> <mspace width="thickmathspace" /> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </math></EquationSource> </Equation> The Hausdorff dimension lower bound result constitutes the main substance of this paper, relying on the Cantor-type subset construction supported by a suitable probability measure and the application of the classical mass distribution principle.

Quantitative analysis of the products of Lüroth digits in arithmetic progressions

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2026年5月9日

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Jing Deng,
Rebecca Smith,
Zhenliang Zhang

摘要

Let [d₁(x), d₂(x), ⋯] be the Lüroth expansion of any real number x ∈ (0, 1]. Fix c ≥ 1 and let Φ: ℕ → (1, ∞) be a positive function. We calculate the Lebesgue measure dichotomy statement (a zero-one law) and the Hausdorff dimension of the following set ${\Lambda _c}(\Phi ): = \left\{ {x \in \left( {0,1} \right]:{d_n}(x){d_{2n}}(x) \cdots {d_{cn}}(x) \geqslant \Phi (n)\;{\rm{for\; infinitely\;many}}\;n \in \mathbb{N}} \right\}.$ $Λ_{c} (Φ) := {x \in (0, 1] : d_{n} (x) d_{2 n} (x) \dots d_{c n} (x) ⩾ Φ (n) f o r i n f i n i t e l y m a n y n \in N} .$ The Hausdorff dimension lower bound result constitutes the main substance of this paper, relying on the Cantor-type subset construction supported by a suitable probability measure and the application of the classical mass distribution principle.