Khintchine dichotomy for self-similar measures
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914194493800448 |
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| author | Bénard, Timothée He, Weikun Zhang, Han |
| author_facet | Bénard, Timothée He, Weikun Zhang, Han |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08061 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Khintchine dichotomy for self-similar measures Bénard, Timothée He, Weikun Zhang, Han Dynamical Systems Number Theory Probability 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. |
| title | Khintchine dichotomy for self-similar measures |
| topic | Dynamical Systems Number Theory Probability |
| url | https://arxiv.org/abs/2409.08061 |