On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions

TL;DR

This paper introduces a new framework for analyzing self-similar measures, proves a formula for their $L^q$-dimensions, and applies it to resolve Furstenberg's intersection conjecture and properties of Bernoulli convolutions.

Contribution

It extends Hochman's entropy approach to $L^q$ norms, providing new tools for understanding self-similar measures and solving longstanding conjectures.

Findings

Resolved Furstenberg's intersection conjecture.

Established Bernoulli convolutions have $L^q$ densities for all finite $q$.

Derived an explicit expression for $L^q$-dimensions of dynamically driven self-similar measures.

Abstract

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-Szemer\'{e}di-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.