# $L^q$ dimensions of self-similar measures, and applications: a survey

## TL;DR

This survey reviews the $L^q$ dimensions of self-similar measures on the real line, providing a simplified proof under exponential separation and discussing applications to Bernoulli convolutions and Cantor set intersections.

## Contribution

It offers a self-contained, simplified proof of the $L^q$ dimension formula for self-similar measures with exponential separation, and reviews key applications.

## Key findings

- Explicit formula for $L^q$ dimensions under exponential separation
- Applications to Bernoulli convolutions
- Insights into intersections of self-similar Cantor sets

## Abstract

We present a self-contained proof of a formula for the $L^q$ dimensions of self-similar measures on the real line under exponential separation (up to the proof of an inverse theorem for the $L^q$ norm of convolutions). This is a special case of a more general result of the author from [Shmerkin, Pablo. On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions. Ann. of Math., 2019], and one of the goals of this survey is to present the ideas in a simpler, but important, setting. We also review some applications of the main result to the study of Bernoulli convolutions and intersections of self-similar Cantor sets.