On the dimension of orthogonal projections of self-similar measures

TL;DR

This paper establishes a criterion involving the group action of an IFS that guarantees the preservation of the dimension of self-similar measures under orthogonal projections, refining previous results and providing sharp conditions for projections to lines and hyperplanes.

Contribution

It introduces a new criterion based on group actions that ensures dimension preservation of self-similar measures under projections, improving upon earlier theorems.

Findings

Dimension preservation under the new criterion

Refinement of Hochman-Shmerkin and Falconer-Jin results

Sharp results for projections onto lines and hyperplanes

Abstract

Let $\nu$ be a self similar measure on $\mathbb{R}^d$, $d\geq 2$, and let $\pi$ be an orthogonal projection onto a $k$-dimensional subspace. We formulate a criterion on the action of the group generated by the orthogonal parts of the IFS on $\pi$, and show that it ensures the dimension of $\pi \nu$ is preserved; this significantly refines previous results by Hochman-Shmerkin (2012) and Falconer-Jin (2014), and is sharp for projections to lines and hyperplanes. A key ingredient in the proof is an application of a restricted projection theorem of Gan-Guo-Wang (2024).