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Absolute continuity of self-similar measures, their projections and convolutions


Authors: Pablo Shmerkin and Boris Solomyak
Journal: Trans. Amer. Math. Soc. 368 (2016), 5125-5151
MSC (2010): Primary 28A78, 28A80; Secondary 37A45, 42A38
Published electronically: June 24, 2015
MathSciNet review: 3456174
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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
Email: pshmerkin@utdt.edu

Boris Solomyak
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350
Email: solomyak@math.washington.edu

DOI: https://doi.org/10.1090/tran6696
Keywords: Absolute continuity, self-similar measures, Hausdorff dimension, convolutions
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
Article copyright: © Copyright 2015 American Mathematical Society