Measures, annuli and dimensions
| Autor: | Buczolich, Zoltán, Seuret, Stéphane |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | Given a Radon probability measure $\mu$ supported in $\mathbb{R}^d$, we are interested in those points $x$ around which the measure is concentrated infinitely many times on thin annuli centered at $x$. Depending on the lower and upper dimension of $\mu$, the metric used in the space and the thinness of the annuli, we obtain results and examples when such points are of $\mu $-measure $0$ or of $\mu$-measure $1$. The measure concentration we study is related to ''bad points'' for the Poincar\'e recurrence theorem and to the first return times to shrinking balls under iteration generated by a weakly Markov dynamical system. The study of thin annuli and spherical averages is also important in many dimension-related problems, including Kakeya-type problems and Falconer's distance set conjecture. Comment: Minor changes compared to the previous version |
| Databáze: | arXiv |
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