Measures, annuli and dimensions - Laboratoire d'Analyse et de Mathématiques Appliquées Accéder directement au contenu
Article Dans Une Revue Mathematische Zeitschrift Année : 2023

Measures, annuli and dimensions

Résumé

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.

Dates et versions

hal-03949007 , version 1 (20-01-2023)

Identifiants

Citer

Zoltán Buczolich, Stéphane Seuret. Measures, annuli and dimensions. Mathematische Zeitschrift, 2023, 303 (4), pp.79. ⟨10.1007/s00209-023-03230-9⟩. ⟨hal-03949007⟩
40 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More