Measures, annuli and dimensions

TL;DR

This paper investigates points where a measure concentrates infinitely often on thin annuli, linking measure dimensions, metric properties, and recurrence phenomena, with implications for dynamical systems and geometric measure theory.

Contribution

It provides new results and examples on measure concentration on thin annuli related to measure dimensions and dynamics, addressing open questions in geometric measure theory.

Findings

Points of measure concentration can have measure zero or one depending on conditions.

Results connect measure concentration to recurrence and dynamical systems.

Examples illustrate the influence of dimensions and metrics on measure behavior.

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 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.