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arXiv 2602.01307math.NTmath.DS
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贾尔尼克型定理对于自相似集

Jarník-type theorem for self-similar sets

Shantou University（汕头大学）；Wuhan University（武汉大学）

Yubin He, Lingmin Liao

备注：
Include several non-trivial self-similar sets whose intersections with the $τ$-approximable set have the desired Hausdorff dimension
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AI总结：

该研究通过建立自相似集与良好逼近数的豪斯多夫维数关系，拓展了均匀和不均匀良好逼近数的维数结果。

摘要：

设$K\subset\mathbb R^d$是一个带有$δ$-阿赫尔福斯正则测度$μ$的紧致子集。对于任何$τ>1/d$和任何\"不均匀\"的向量$\boldsymbolθ\in\mathbb R^d$，令$W_d(ψ_τ,\boldsymbolθ)$表示$(ψ_τ,\boldsymbolθ)$-良好逼近数的集合，其中$ψ_τ(q)=q^{-τ}$。假设对$K$与\"有理\"向量$(\mathbf p+\boldsymbolθ)/q$的邻域的交集有一个局部估计，我们建立了$K\cap W_d(ψ_τ,\boldsymbolθ)$的豪斯多夫维数的精确上界，并且在$τ$低于某个阈值时提供了一些非平凡的下界。其中一个下界在一维均匀情况（$d=1$，$θ=0$）下对于足够厚的自相似集$K$变得精确，并且$K\cap W_1(ψ_τ,0)$具有$(δ+\frac{2}{1+τ}-1)$-豪斯多夫测度。这些结果有几种应用：(1) 均匀非常良好逼近数的集合在强不可约自相似集$\mathbb R^d$中具有完整的豪斯多夫维数，扩展了陈[arXiv:2510.17096]最近的结果；(2) 不均匀非常良好逼近数的集合在足够厚的缺失数字集$\mathbb R$中具有完整的豪斯多夫维数，肯定回答了尤[arXiv:2101.05910]提出的问题。我们的应用基于尤[arXiv:2101.05910]和本德、何和张[arXiv:2508.09076]的奠基性工作。我们还提供了一些非平凡缺失数字集$K\subset[0,1]^d$，其与$W_d(ψ_τ,0)$的交集具有完整的$(δ+\frac{1+d}{1+τ}-d)$-豪斯多夫测度。

英文摘要：

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $δ$-Ahlfors regular measure $μ$. For any $τ>1/d$ and any ``inhomogeneous'' vector $\boldsymbolθ\in\mathbb R^d$, let $W_d(ψ_τ,\boldsymbolθ)$ denote the set of $(ψ_τ,\boldsymbolθ)$-well approximable numbers, where $ψ_τ(q)=q^{-τ}$. Assuming a local estimate for the $μ$-measure of the intersections of $K$ with the neighborhoods of ``rational'' vectors $(\mathbf p+\boldsymbolθ)/q$, we establish a sharp upper bound for the Hausdorff dimension of $K\cap W_d(ψ_τ,\boldsymbolθ)$, together with some nontrivial lower bounds when $τ$ is below a certain threshold. One of the lower bounds becomes sharp in the one-dimensional homogeneous case ($d=1$, $θ=0$) for a class of sufficiently thick self-similar sets $K$, and moreover $K\cap W_1(ψ_τ,0)$ has full $(δ+\frac{2}{1+τ}-1)$-Hausdorff measure. These results have several applications: (1) the set of homogeneous very well approximable numbers has full Hausdorff dimension within strongly irreducible self-similar sets in $\mathbb R^d$, extending a recent result of Chen [arXiv:2510.17096]; (2) the set of inhomogeneous very well approximable numbers has full Hausdorff dimension within sufficiently thick missing digits sets in $\mathbb R$, affirmatively answering a question posed by Yu [arXiv:2101.05910]. Our applications build on the seminal works of Yu [arXiv:2101.05910] and Bénard, He and Zhang [arXiv:2508.09076]. We also provide some non-trivial missing digits set $K\subset[0,1]^d$ whose intersection with $W_d(ψ_τ,0)$ has full $(δ+\frac{1+d}{1+τ}-d)$-Hausdorff measure.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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