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

TL;DR

This paper investigates the conditions under which the convolution of self-conformal measures in  attains a specific dimension formula, highlighting the roles of non-linearity and support on smooth hypersurfaces.

Contribution

It establishes sufficient geometric and algebraic conditions for the dimension of convolutions of self-conformal measures, extending understanding of measure interactions in .

Findings

Dimension formula holds under total non-linearity and non-support on smooth hypersurfaces.

Provides algebraic conditions for measures that are not totally non-linear.

Combines recent scaling scenery results with a Marstrand-type projection theorem.

Abstract

Let $\mu$ be a self-conformal measure on $\mathbb{R}^d$. In this note we establish conditions for $\mu$ under which $\dim(\mu*\nu) = \min\lbrace d,\dim\mu+\dim\nu\rbrace$ holds when $\nu$ is any Ahlfors-regular or self-conformal measure on $\mathbb{R}^d$. Our main result states the following sufficient condition: $\mu$ 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 L\'opez and Moreira.