On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions

By Pablo Shmerkin. Annals of Mathematics, 2016

183 citations recorded by the source corpus.

We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the $L^q$-dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of $\times p$ and $\times q$-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an $L^q$ density for all finite $q$, outside of a zero-dimensional set of exceptions. The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to $L^q$ norms, and likewise relies on an inverse theorem for the decay of $L^q$ norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemeredi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f8f9ef810fafad527dbc7c4c2f424cf4fb957639

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Citations in the corpus

A proof of Furstenberg's conjecture on the intersections of ⨯p- and ⨯q-invariant sets (2016, methodology): “Recently, there has been considerable interest in the study of projections of dynamically defined Cantor sets, see for instance the survey paper of Shmerkin [34] and the references therein for more details.”
On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane (2021, background): “Furstenberg [10] on intersections of Cantor sets defined in terms of expansions to bases 2, 3 (the conjecture that motivated this connection has since been resolved by the second author [27] and, independently, by M.”