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arXiv · 2409.08061

Khintchine dichotomy for self-similar measures

Timothée Bénard·Weikun He·Han Zhang
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Abstract

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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Timothée Bénard, Weikun He, Han Zhang. 2024-09-12. Khintchine dichotomy for self-similar measures. https://arxiv.org/abs/2409.08061

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