On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions

• Published in: Geometric and functional analysis Vol. 24; no. 3; pp. 946 - 958
• Main Author: Shmerkin, Pablo
• Format: Journal Article
• Language: English
• Published: Basel Springer Basel 01.06.2014
• Subjects: Analysis; Mathematics; Mathematics and Statistics; 28A80; Bernoulli convolutions; Primary 28A78; self-similar measures; Secondary 37A45; hausdorff dimension

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,...