Absolute continuity of projections and dimension conservation for self-similar sets and measures

Abstract

Let $A\subset {\mathbb{R}}^d$, $d\ge 3$, be the attractor of a self-similar IFS whose rotations generate a dense subgroup of $\mathrm{SO}(d)$. For every $1\le k<{\dim }_{\mathrm{H}}A$ and $V\in \mathrm{Gr}(d,k)$, we prove that ${\pi }_{V}A$ has positive $k$-dimensional Lebesgue measure uniformly in $V$, and ${\dim }_{\mathrm{H}}(A\cap {\pi }_V^{-1}\{y\})={\dim }_{\mathrm{H}}A-k$ for Lebesgue-almost every $y\in {\pi }_{V}A$. Moreover, if $\mu$ is a self-similar measure for such an IFS with $I_t(\mu )<\infty$ for some $k<t<d$, then every ${\mu }_V=({\pi }_V{)}_{\ast }\mu$ is equivalent to the $k$-dimensional Lebesgue measure restricted to ${\pi }_{V}A$, with density $f_V$ satisfying ${\sup }_V{\int }_{\{f_V>M\}}{f}_{V}d{\mathcal{L}}_V^k\lesssim e^{-c(\log M{)}^{1/3}}$. Under strong separation, the conditional measures on the fibres are exact dimensional of dimension ${\dim }_{\mathrm{H}}\mu -k$. The proof uses self-similarity to iterate $L^1$ norms of differences of the Gaussian-smoothing of ${\mu }_V$ between scales. Unlike $L^q$, $q>1$, the $L^1$ norm does not grow under rescaling, allowing Varjú's $L^2$ weak gap theorem under dense support condition to be used, and avoiding to use the much stronger uniform spectral gap assumption.

arXiv comment30 pages; version 2: included unequal contraction ratios case for self-similar measures, updated the exposition, references and introduction