自相似度量的钦钦二分法

arXiv - MATH - Probability Pub Date : 2024-09-12 DOI:arxiv-2409.08061

Timothée Bénard, Weikun He, Han Zhang

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引用次数: 0

摘要

我们为实线上的所有自相似概率度量建立了Khintchine定理的类似物。当把这一定理应用于中三康托尔集上的豪斯多夫度量时，它已经是一个新结果，并为马勒的一个老问题提供了答案。证明包括在$\text{SL}_{2}(\mathbb{R})/\text{SL}_{2}(\mathbb{Z})$上的扩展上三角随机游走法则中显示有效的等差数列，这是一个具有独立意义的结果。

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Khintchine dichotomy for self-similar measures

We establish the analogue of Khintchine's theorem for all self-similar probability measures on the real line. When specified to the case of the Hausdorff measure on the middle-thirds Cantor set, the result is already new and provides an answer to an old question of Mahler. The proof consists in showing effective equidistribution in law of expanding upper-triangular random walks on $\text{SL}_{2}(\mathbb{R})/\text{SL}_{2}(\mathbb{Z})$, a result of independent interest.

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arXiv - MATH - Probability

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