On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions
We prove that the set of exceptional λ∈ (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 Erdös, Kahane, Solomyak, Peres...
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2014
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_1016443X_v24_n3_p946_Shmerkin http://hdl.handle.net/20.500.12110/paper_1016443X_v24_n3_p946_Shmerkin |
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| Sumario: | We prove that the set of exceptional λ∈ (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 Erdös, 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. © 2014 Springer Basel. |
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