Measures, annuli and dimensions

Abstract

Given a Radon probability measure $$\mu $$ μ supported in $${\mathbb {R}}^d$$ 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 $$\mu $$ μ , the metric used in the space and the thinness of the annuli, we obtain results and examples when such points are of $$\mu $$ μ -measure 0 or of $$\mu $$ μ -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.