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

Shmerkin, Pablo; Solomyak, Boris

doi:10.1090/tran6696

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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