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.
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