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Abstract:

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.

Registro:

Documento: Artículo
Título:On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions
Autor:Shmerkin, P.
Filiación:Department of Mathematics and Statistics, Torcuato Di Tella University, Av. Figueroa Alcorta 7350, 1428 Buenos Aires, Argentina
Palabras clave:28A80; Bernoulli convolutions; hausdorff dimension; Primary 28A78; Secondary 37A45; self-similar measures
Año:2014
Volumen:24
Número:3
Página de inicio:946
Página de fin:958
DOI: http://dx.doi.org/10.1007/s00039-014-0285-4
Título revista:Geometric and Functional Analysis
Título revista abreviado:Geom. Funct. Anal.
ISSN:1016443X
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_1016443X_v24_n3_p946_Shmerkin

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Citas:

---------- APA ----------
(2014) . On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions. Geometric and Functional Analysis, 24(3), 946-958.
http://dx.doi.org/10.1007/s00039-014-0285-4
---------- CHICAGO ----------
Shmerkin, P. "On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions" . Geometric and Functional Analysis 24, no. 3 (2014) : 946-958.
http://dx.doi.org/10.1007/s00039-014-0285-4
---------- MLA ----------
Shmerkin, P. "On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions" . Geometric and Functional Analysis, vol. 24, no. 3, 2014, pp. 946-958.
http://dx.doi.org/10.1007/s00039-014-0285-4
---------- VANCOUVER ----------
Shmerkin, P. On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions. Geom. Funct. Anal. 2014;24(3):946-958.
http://dx.doi.org/10.1007/s00039-014-0285-4