AI 中文总结
研究沿整数康托集的遍历平均的逐点收敛，通过证明相关遍历平均在特定条件下几乎处处收敛，解决了康托测度在自相似尺度下的稀疏微分问题。
AI 中文摘要
设\(d\geq3\)，\(D\subsetneq\{0,1,\dots,d - 1\}\)，\(|D|\geq2\)且\(0\in D\)为有限字母表，定义整数康托集\(\mathcal{C}:=\mathcal{C}_{D}:=\bigcup_{J\geq0}\{\sum_{j = 0}^J a_j d^j : a_j\in D\}\)。我们证明对于任何\(\sigma -\)有限保测系统\((X,\mu,T)\)以及任何\(f\in L^p(X)\)，\(2\leq p\lt\infty\)，遍历平均\(\frac{1}{|\mathcal{C}_N|}\sum_{n\in\mathcal{C}_N}f(T^n x)\)，\(\mathcal{C}_N:=\mathcal{C}\cap\{1,2,\dots,N\}\)几乎处处\(\mu -\)收敛。通过重新缩放，这解决了康托测度在自相似尺度下的稀疏微分问题：若\(\mathcal{C}':=\{\sum_{j\geq1}a_j d^{-j}:a_j\in D\}\subset[0,1]\)是实变量康托集，\(\nu\)表示其自然测度，则对于任何\(f\in L^2_{\text{loc}}(\mathbb{R})\)，\(\lim_{k\rightarrow\infty}\int f(x - d^{-k}t)d\nu(t)=f(x)\)几乎处处勒贝格收敛。
英文摘要
Let $d \geq 3$, \[ D \subsetneq \{ 0,1,\dots,d-1\}, \qquad |D| \geq 2, \ 0 \in D \] be a finite alphabet, and define the integer Cantor set \begin{align} \mathcal{C} := \mathcal{C}_{D} := \bigcup_{J \geq 0} \Big\{ \sum_{j =0}^J a_j d^j : a_j \in D \Big\}. \end{align} We prove that for any $σ$-finite measure-preserving system, $(X,μ,T)$, and any $f \in L^p(X)$, $2\leq p<\infty$, the ergodic averages \begin{align} \frac{1}{|\mathcal{C}_N|} \sum_{n \in \mathcal{C}_N } f(T^n x), \qquad \mathcal{C}_N := \mathcal{C} \cap \{1,2,\dots,N \} \end{align} converge $μ$-almost everywhere. By rescaling, this allows us to resolve the question of lacunary differentiation of Cantor measures at self-similar scales: if \begin{align} \mathcal{C}' := \Big\{ \sum_{j \geq 1} a_j d^{-j} : a_j \in D \Big\} \subset [0,1] \end{align} is a real-variable Cantor set, and $ν$ denotes its natural measure, then we prove that \begin{align} \lim_{k \to \infty} \int f(x-d^{-k} t) \ dν(t) = f(x) \end{align} Lebesgue almost-everywhere for any $f \in L^2_{\text{loc}}(\mathbb{R})$.
