On dissonance of self-conformal measures in $\mathbb{R}^d$

TL;DR

The paper investigates when the dimension of the convolution $\dim(\mu*\nu)$ attains the Marstrand-type bound $\min\{d,\dim\mu+\dim\nu\}$ for a self-conformal measure $\mu$ and a broad class of measures $\nu$ on $\mathbb{R}^d$. It develops a three-part strategy based on tangent-distributions (scaling scenery), invariance under conformal linear maps, and a projection theorem for product sets to transfer dimension information under addition. The main contribution is a sufficient condition: if $\mu$ is totally non-linear and not supported on a smooth hypersurface, then $\dim(\mu*\nu)=\min\{d,\dim\mu+\dim\nu\}$ for any tangent-regular $\nu$ (which includes Ahlfors-regular and self-conformal measures), complemented by algebraic-condition variants that replace non-linearity in higher dimensions. These results extend the understanding of dissonance beyond the line and plane, provide dimension preservation under projections, and advance the use of scaling scenery and modern projection theorems in the study of self-conformal measures.

Abstract

Let $μ$ be a self-conformal measure on $\mathbb{R}^d$. In this note we establish conditions for $μ$ under which $\dim(μ*ν) = \min\lbrace d,\dimμ+\dimν\rbrace$ holds when $ν$ is any Ahlfors-regular or self-conformal measure on $\mathbb{R}^d$. Our main result states the following sufficient condition: $μ$ is totally non-linear and not supported on a smooth hypersurface. We also establish sufficient (likely non-sharp) algebraic conditions for self-conformal measures which are not totally non-linear. The proofs combine recent results on scaling sceneries of self-conformal measures with a Marstrand-type projection theorem for product sets due to López and Moreira.

Theorems & Definitions (45)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Corollary 1.7
• Corollary 1.8
• Theorem 1.9
• Theorem 1.10