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Absolute continuity of Bernoulli convolutions for algebraic parameters


Author: Péter P. Varjú
Journal: J. Amer. Math. Soc. 32 (2019), 351-397
MSC (2010): Primary 28A80, 42A85
DOI: https://doi.org/10.1090/jams/916
Published electronically: January 22, 2019
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Abstract: We prove that Bernoulli convolutions $ \mu _\lambda $ are absolutely continuous provided the parameter $ \lambda $ is an algebraic number sufficiently close to $ 1$ depending on the Mahler measure of $ \lambda $.


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Additional Information

Péter P. Varjú
Affiliation: Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
Email: pv270@dpmms.cam.ac.uk

DOI: https://doi.org/10.1090/jams/916
Keywords: Bernoulli convolution, self-similar measure, absolute continuity, Mahler measure
Received by editor(s): August 16, 2016
Received by editor(s) in revised form: May 15, 2017, and June 27, 2018
Published electronically: January 22, 2019
Additional Notes: The author gratefully acknowledges the support of the Royal Society.
Article copyright: © Copyright 2019 American Mathematical Society