Given <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha >1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, every real number <InlineEquation ID="IEq5"> <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> can be expanded into a power-<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-decaying Gauss-like expansion <Equation ID="Equ37"> <EquationSource Format="TEX">$\begin{aligned} x=\sum \limits _{n=1}^{\infty }\frac{(\alpha -1)^{n-1}}{\alpha ^{d_1(x)+\cdots +d_n(x)}},\ \ d_i(x)\in \mathbb {N}. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>x</mi> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </munderover> <mfrac> <msup> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>α</mi> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>d</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> </mfrac> <mo>,</mo> <mspace width="4pt" /> <mspace width="4pt" /> <msub> <mi>d</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Denoted by <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$l_n(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>l</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> the longest run of the same symbol in the first n digits of x. This paper is concerned with the asymptotic behavior of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$l_n(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>l</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We prove that the rate of growth of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$l_n(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>l</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is nearly <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\log _{\frac{\alpha }{\alpha -1}} n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>log</mo> <mfrac> <mi>α</mi> <mrow> <mi>α</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </msub> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, and obtain that the exceptional set of points for which <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$l_n(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>l</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> grows logarithmically is of full Hausdorff dimension. We also establish the Hausdorff dimension of the set <Equation ID="Equ131"> <MediaObject ID="MO1"> <ImageObject Color="BlackWhite" FileRef="MediaObjects/40840_2026_2046_Equ131_HTML.png" Format="PNG" Height="100" Rendition="HTML" Resolution="300" Type="Linedraw" Width="1010" /> </MediaObject> </Equation>for any <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$0\le c\le d\le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>c</mi> <mo>≤</mo> <mi>d</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.

On the Maximal Run-Length Function in the Power-$\alpha $-Decaying Gauss-Like Expansion

期刊论文
发表日期：

2026年1月28日

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Lingling Huang,
Yuanhong Chen

摘要

Given $\alpha >1$ $α > 1$ , every real number $x\in (0,1]$ $x \in (0, 1]$ can be expanded into a power- $\alpha $ $α$ -decaying Gauss-like expansion $\begin{aligned} x=\sum \limits _{n=1}^{\infty }\frac{(\alpha -1)^{n-1}}{\alpha ^{d_1(x)+\cdots +d_n(x)}},\ \ d_i(x)\in \mathbb {N}. \end{aligned}$ $\begin{matrix} x = \sum_{n = 1}^{\infty} \frac{{(α - 1)}^{n - 1}}{α^{d_{1} (x) + \dots + d_{n} (x)}}, d_{i} (x) \in N . \end{matrix}$ Denoted by $l_n(x)$ $l_{n} (x)$ the longest run of the same symbol in the first n digits of x. This paper is concerned with the asymptotic behavior of $l_n(x)$ $l_{n} (x)$ . We prove that the rate of growth of $l_n(x)$ $l_{n} (x)$ is nearly $\log _{\frac{\alpha }{\alpha -1}} n$ ${log}_{\frac{α}{α - 1}} n$ , and obtain that the exceptional set of points for which $l_n(x)$ $l_{n} (x)$ grows logarithmically is of full Hausdorff dimension. We also establish the Hausdorff dimension of the set for any $0\le c\le d\le 1$ $0 \leq c \leq d \leq 1$ .

On the Maximal Run-Length Function in the Power-\(\alpha \)-Decaying Gauss-Like Expansion

摘要