On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions
We prove that the set of exceptional λ∈(1/2,1)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...
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2014 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/35642 |
| Acceso en línea: | http://hdl.handle.net/11336/35642 |
| Access Level: | acceso abierto |
| Palabra clave: | Bernoulli Convolutions Self-Similarity Absolute Continuity https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | We prove that the set of exceptional λ∈(1/2,1)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. |
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