ℝᵈ中自相似集上τ-可逼近点的豪斯多夫维数
Hausdorff dimension of $τ$-approximable points on self-similar sets in $\mathbb R^d$
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中文总结 AI 辅助
该研究针对ℝᵈ中满足开集条件的有限强不可约迭代函数系生成的非单点自相似集，确定了其上τ-可逼近点集与该自相似集交集的豪斯多夫维数，在一维中间三分康托集情形为Bugeaud-Durand猜想公式提供了支撑。
中文摘要 AI 辅助
设d≥1，K是ℝᵈ中由满足开集条件的有限强不可约迭代函数系生成的非单点自相似集，δ=dim_H K。对τ>1/d，定义W_d(τ)为ℝᵈ中满足存在无穷多组(p,q)∈ℤᵈ×ℕ使得|q𝐱-p|<q^(-τ)的点集。我们证明存在ε_K>0，对所有1/d<τ<1/d+ε_K，有ℋ^(s(τ))(K∩W_d(τ))=∞，其中s(τ)=δ+(d+1)/(1+τ)-d，进而得dim_H(K∩W_d(τ))=δ+(d+1)/(1+τ)-d。在一维情形下，针对中间三分康托集，这为τ>1且足够接近1时的Bugeaud-Durand猜想公式提供了支撑。
英文摘要
Let $d\geq 1$. Let $K\subset\mathbb{R}^d$ be a non-singleton self-similar set generated by a finite strongly irreducible iterated function system satisfying the open set condition, and let $δ=\dim_{\mathrm H} K$. For $τ>1/d$, set \[ W_d(τ) = \left\{ \mathbf{x}\in\mathbb{R}^d: |q\mathbf{x}-\mathbf{p}|<q^{-τ} \text{ for infinitely many }(\mathbf{p},q)\in\mathbb{Z}^d\times\mathbb{N} \right\}. \] We prove that there exists $\varepsilon_K>0$ such that, for every $1/d<τ<1/d+\varepsilon_K$, \[ \mathcal{H}^{s(τ)}(K\cap W_d(τ))=\infty, \qquad\text{with } s(τ):=δ+\frac{d+1}{1+τ}-d, \] and consequently \[ \dim_{\mathrm H}(K\cap W_d(τ)) = δ+\frac{d+1}{1+τ}-d. \] In dimension one, specializing to the middle-third Cantor set, this establishes the Bugeaud--Durand conjectural formula for $τ>1$ sufficiently close to $1$.
