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Dynamics Lunch: Lauritz Streck (Cambridge) - Maximal Dimension for Bernoulli Convolutions with Transcendental Parameter | Einstein Institute of Mathematics

Dynamics Lunch: Lauritz Streck (Cambridge) - Maximal Dimension for Bernoulli Convolutions with Transcendental Parameter

Date: 
Tue, 05/04/202212:00-13:00
The Bernoulli convolution measure with parameter a<1 is the law of the
power series in a with random, independent signs attaining
plus and minus with equal probability. In 1953, Erdős proved that the
measure is not absolutely continuous for certain algebraic a, the Pisot
numbers, and conjectured that these are the only exceptions. This, and
the related question of when the Hausdorff dimension is 1, has sparked
considerable interest and has seen a lot of activity. For example, it is
known that for Lebesgue almost all 0.5<a<1 the measure is absolutely
continuous (Solomyak) and that the Hausdorff dimension of the parameters
with dimension drop of the measure occurring is zero (Hochman). However,
there were still only few explicit values for which the measure is known
to be absolutely continuous or to have dimension 1.
We will present the proof of Peter Varju from 2019 that the measure has
dimension 1 for all transcendental a. The proof combines results of
Varju and Breuillard on entropy estimates with the ideas of many
previous authors, so the talk will organically include many of the main
ideas in the field.