On dissonance and orthogonal projections of self-conformal measures

Abstract

Let $\mu$ be a self-conformal measure on ${\mathbb{R}}^d$. We establish conditions for $\mu$ under which $\dim (\mu \ast \nu )=\min \{d,\dim \mu +\dim \nu \}$ 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. In addition, we show that $\dim \mu \circ {\pi }^{-1}=\min \{k,\dim \mu \}$ for every ortohogonal projection $\pi :{\mathbb{R}}^d\to {\mathbb{R}}^k$, $0<k<d$, when either $d=2$ and $\mu$ is not self-similar and not supported on a line, or $d\ge 3$ and $\mu$ is totally non-linear and not supported on a smooth hypersurface.

arXiv comment26 pages. v2: Expanded on the application to orthogonal projections, including new Theorem 1.8, and several minor fixes and clarifications