<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$T_\beta ~(\beta >2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mi>β</mi> </msub> <mspace width="3.33333pt" /> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo>></mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-transformation on (0, 1]. Fix some <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$x_0\in [0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>∈</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> whose <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-expansion does not include the characters 0 and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lceil \beta \rceil -1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>⌈</mo> <mi>β</mi> <mo>⌉</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, given a nonnegative real number <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\hat{v}}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </math></EquationSource> </InlineEquation>, we compute the Hausdorff dimension of the set of all real numbers <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$x\in (0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> with the property that, for every sufficiently large integer <i>N</i>, there is an integer <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n\in [1,N]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> such that the distance between <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$T_\beta ^nx$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>T</mi> <mi>β</mi> <mi>n</mi> </msubsup> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$x_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> is at most equal to <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\beta ^{-N{\hat{v}}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>β</mi> <mrow> <mo>-</mo> <mi>N</mi> <mover accent="true"> <mi>v</mi> <mo stretchy="false">^</mo> </mover> </mrow> </msup> </math></EquationSource> </InlineEquation>. This work generalizes the result of Bugeaud and Liao [<CitationRef CitationID="CR9">9</CitationRef>] where only <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$x_0=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> was taken into consideration.</p>

Uniform Diophantine approximation related to beta-transformations

期刊论文
发表日期：

2025年10月21日

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Wanlou Wu,
Yingqing Zhang

摘要

Let $T_\beta ~(\beta >2)$ $T_{β} (β > 2)$ be the $\beta $ $β$ -transformation on (0, 1]. Fix some $x_0\in [0,1]$ $x_{0} \in [0, 1]$ whose $\beta $ $β$ -expansion does not include the characters 0 and $\lceil \beta \rceil -1$ $⌈ β ⌉ - 1$ , given a nonnegative real number ${\hat{v}}$ $\hat{v}$ , we compute the Hausdorff dimension of the set of all real numbers $x\in (0,1]$ $x \in (0, 1]$ with the property that, for every sufficiently large integer N, there is an integer $n\in [1,N]$ $n \in [1, N]$ such that the distance between $T_\beta ^nx$ $T_{β}^{n} x$ and $x_0$ $x_{0}$ is at most equal to $\beta ^{-N{\hat{v}}}$ $β^{- N \hat{v}}$ . This work generalizes the result of Bugeaud and Liao [9] where only $x_0=0$ $x_{0} = 0$ was taken into consideration.