$L^q$ dimensions and projections of random measures

TL;DR

This paper demonstrates that for a class of random measures on the plane, their $L^q$ dimensions are preserved under all orthogonal projections, extending known results from Hausdorff dimension to $L^q$ dimensions.

Contribution

It establishes the preservation of $L^q$ dimensions under projections for certain random measures, including self-similar and fractal measures, and extends results to convolutions.

Findings

$L^q$ dimensions are preserved under projections for the studied measures.

The results include self-similar and fractal measures as special cases.

Extension of known results from Hausdorff to $L^q$ dimensions.

Abstract

We prove preservation of $L^q$ dimensions (for $1<q\le 2$) under all orthogonal projections for a class of random measures on the plane, which includes (deterministic) homogeneous self-similar measures and a well-known family of measures supported on $1$-variable fractals as special cases. We prove a similar result for certain convolutions, extending a result of Nazarov, Peres and Shmerkin. Recently many related results have been obtained for Hausdorff dimension, but much less is known for $L^q$ dimensions.