Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions

By Yuval Peres, Wilhelm Schlag. 2000

254 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d1ee5ecac7519d91fb90e407ef3da498e8d49ad9

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In Applied Mathematics

In Geometry and Topology

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Citations in the corpus

On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions (2013, background): “parameter in a given parametrized family, the expected behavior holds (in both regimes), and in many cases of interest it is was shown by Peres and Schlag [19] that the dimension of the exceptional set is less than full.”
On self-similar sets with overlaps and inverse theorems for entropy (2012, methodology): “The study of parametric families of general sets and measures is classical; examples include the projection theorems of Besicovitch and Marstrand and more recent results like those of Peres-Schlag [24] and Bourgain [3].”
Refinable functions with non-integer dilations (2007, background): “In the generic setting Solomyak [16] proved that for almost all dilations λ ∈ (1,2) the corresponding Bernoulli convolution is in L1, and more recently, Peres and Schlag [13] proved that the Fourier transform of the Bernoulli convolution has uniform decay at infinity for almost all dilations λ ∈ (1,”
On the dimension and smoothness of radial projections (2017, background): “One can check that the family of projections {πx}x∈Rd\sptμ fits in the generalised projections framework of Peres and Schlag [16], and indeed Theorem 7.”
On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions (2016, background): “In the same article [36], Peres and Schlag proved that for every ε > 0, there is an explicit δ > 0 such that dimH{λ ∈ (1/2 + ε, 1) : νλ does not have an L(2) density } ≤ 1− δ.”
Projection and slicing theorems in Heisenberg groups (2012, unknown): “…in Mathematics 231 (2012) 569–604 Acknowledgments.............................................................................................................603 References .......................................................................................................................603”
Recent progress on Bernoulli convolutions (2016, background): “ulli convolution νλ is absolutely continuous and has density in L2 for almost every λ ∈ (1/2,1). Solomyak’s original proof has been simpliﬁed in [49]. The result has been improved by Peres and Schlag [47], who proved that the density also has fractional derivatives in L2 and they gave estimates for ”