On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions

By Pablo Shmerkin. 2013

124 citations recorded by the source corpus.

We prove that the set of exceptional $${\lambda\in (1/2,1)}$$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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/09f744b78c9c3243e882a50f136179791079c119

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

On self-similar sets with overlaps and inverse theorems for entropy (2012, unknown): “For instance, Theorem (Shmerkin [31]).”
On the modulus of continuity for spectral measures in substitution dynamics (2013, unknown): “Having dimension one for a measure falls just short of absolute continuity, but even more recently Shmerkin [37] combined Hochman’s Theorem with the Erdős-Kahane result mentioned above to obtain absolute continuity outside a zero-dimensional set of exceptions.”
On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions (2016, background): “The main ideas in this section are borrowed from [42, 45]; the reason we improve upon existing results is that Theorem 1.”
On the dimension of Bernoulli convolutions for all transcendental parameters (2018, background): “The reader interested in Question 1 is also recommended to consult the recent papers [18, 19, 23].”
Absolute continuity of Bernoulli convolutions for algebraic parameters (2016, methodology): “Unfortunately, the methods of [20] do not seem to apply for specific parameters, hence we need a different method.”
Weak convergence and spectrality of infinite convolutions (2022, background): “These studies revealed connections with harmonic analysis, the theory of algebraic numbers, dynamical systems, and fractal geometry [9, 28, 48, 49].”
Recent progress on Bernoulli convolutions (2016, background): “Shmerkin [55] achieved the following important result on absolute continuity of νλ for typical parameters.”