Absolute continuity of self-similar measures, their projections and convolutions

By Pablo Shmerkin, Boris Solomyak. 2014

78 citations recorded by the source corpus.

We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small - of co-dimension at least one in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continuity of the convolution of two measures. As a concrete application, we obtain a very strong version of Marstrand's projection theorem for planar self-similar sets.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4744e83f836baf5f432f35abf6f3b8438d8b135a

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems (2016, result): “sion of projections of dynamically deﬁned sets and measures has received a great deal of attention in the last decade, and the situation is fairly well understood in the selfsimilar setting, see e.g. [32, 33] and references there. In the self-aﬃne case, some results were obtained in the carpet case ”
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.”
Recent progress on Bernoulli convolutions (2016, background): “yields not only that νλ is absolutely continuous (if λ is outside the exceptional set) but also that it has some fractional derivatives in Lp for some p > 1 (depending on λ). Shmerkin and Solomyak [56], [57] extended these ideas to more general self-similar measures. We conclude with the followin”