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On the exceptional set for absolute continuity of Bernoulli convolutions

Shmerkin, P (2013) On the exceptional set for absolute continuity of Bernoulli convolutions Geom. Funct. Anal. 24 (2014), no. 3, 946--958.

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Abstract

We prove that the set of exceptional $\lambda\in (1/2,1)$ such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the bias. This improves previous results by Erd\"os, Kahane, Solomyak, Peres and Schlag, and Hochman. A theorem of this kind is also obtained for convolutions of homogeneous self-similar measures. The proofs are very short, and rely on old and new results on the dimensions of self-similar measures and their convolutions, and the decay of their Fourier transform.

Item Type: Article
Authors :
NameEmailORCID
Shmerkin, Pp.shmerkin@surrey.ac.ukUNSPECIFIED
Date : 16 March 2013
Uncontrolled Keywords : math.DS, math.DS, math.CA, 28A78, 28A80, 37A45
Related URLs :
Depositing User : Symplectic Elements
Date Deposited : 17 May 2017 12:46
Last Modified : 17 May 2017 15:06
URI: http://epubs.surrey.ac.uk/id/eprint/836683

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