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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Absolute continuity of self-similar measures, their projections and convolutions
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by Pablo Shmerkin and Boris Solomyak PDF
Trans. Amer. Math. Soc. 368 (2016), 5125-5151 Request permission

Abstract:

We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small—of co-dimension at least 1 in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continuity of the convolution of two measures. As a concrete application, we obtain a very strong version of Marstrand’s projection theorem for planar self-similar sets.
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Additional Information
  • Pablo Shmerkin
  • Affiliation: Department of Mathematics and Statistics, Torcuato di Tella University, and CONICET, Av. Figueroa Alcorta 7350 (1425), Buenos Aires, Argentina
  • MR Author ID: 781925
  • Email: pshmerkin@utdt.edu
  • Boris Solomyak
  • Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350
  • MR Author ID: 209793
  • Email: solomyak@math.washington.edu
  • Received by editor(s): June 29, 2014
  • Published electronically: June 24, 2015
  • Additional Notes: The first author was supported in part by Project PICT 2011-0436 (ANPCyT)
    The second author was supported in part by NSF grant DMS-0968879 and by the Forschheimer Fellowship and grant ERC AdG 267259 at the Hebrew University of Jerusalem
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 5125-5151
  • MSC (2010): Primary 28A78, 28A80; Secondary 37A45, 42A38
  • DOI: https://doi.org/10.1090/tran6696
  • MathSciNet review: 3456174