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

Abstract

Let $A\subset\mathbb{R}^d$, $d\geq3$, be the attractor of a self-similar IFS whose rotations generate a dense subgroup of $\mathrm{SO}(d)$. For every $1\leq k<\dim_{\mathrm{H}} A$ and $V\in\mathrm{Gr}(d,k)$, we prove that $\pi_VA$ 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 $y\in\pi_VA$. Moreover, if $\mu$ is a self-similar measure for such an IFS with $I_t(\mu)<\infty$ for some $k<t<d$, then every $\mu_V=(\pi_V)_*\mu$ is equivalent to the $k$-dimensional Lebesgue measure restricted to $\pi_VA$, 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 Varj\'u's $L^2$ weak gap theorem under dense support condition to be used, and avoiding to use the much stronger uniform spectral gap assumption.