Measures, annuli and dimensions

Description of video

Date: 5/24/23
Speaker :Buczolich Zoltán


    Abstract: 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 Poincaré 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.

    This talk is based on a joint paper with Stéphane Seuret.