Geodesic
On Furstenberg’s intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions
Pablo Shmerkin1
1 Department of Mathematics and Statistics, Torcuato Di Tella University, and CONICET, Buenos Aires, Argentina
Annals of mathematics, Tome 189 (2019) no. 2, pp. 319-391

Voir la notice de l'article provenant de la source Annals of Mathematics website

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.

DOI : 10.4007/annals.2019.189.2.1

Pablo Shmerkin 1

1 Department of Mathematics and Statistics, Torcuato Di Tella University, and CONICET, Buenos Aires, Argentina 
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Pablo Shmerkin. On Furstenberg’s intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions. Annals of mathematics, Tome 189 (2019) no. 2, pp. 319-391. doi: 10.4007/annals.2019.189.2.1

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