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- W3212078012 abstract "Abstract Given a Radon probability measure $$mu $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>μ</mml:mi> </mml:math> supported in $${mathbb {R}}^d$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , 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 $$mu $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>μ</mml:mi> </mml:math> , the metric used in the space and the thinness of the annuli, we obtain results and examples when such points are of $$mu $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>μ</mml:mi> </mml:math> -measure 0 or of $$mu $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>μ</mml:mi> </mml:math> -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." @default.
- W3212078012 created "2021-11-22" @default.
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- W3212078012 date "2023-02-26" @default.
- W3212078012 modified "2023-09-30" @default.
- W3212078012 title "Measures, annuli and dimensions" @default.
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- W3212078012 doi "https://doi.org/10.1007/s00209-023-03230-9" @default.
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