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

Abstract: Let $A\subset\mathbb{R}<sup>d$, $d\ge3$, be a self-similar set whose defining rotations generate a dense subgroup of $\mathrm{SO}(d)$. For every integer $1\le k&lt;\dim_{\mathrm{H}} A$, we prove that its orthogonal projections $π<em>V A$ have uniformly positive $k$-dimensional Lebesgue measure, and that their fibres have Hausdorff dimension $\dim</em>{\mathrm{H}} A-k$ at Lebesgue-almost every point of the projected image. This follows from an absolute-continuity theorem for equicontractive self-similar measures $μ$ with the same rotation hypothesis and finite $t$-energy for some $t&gt;k$. Every projected measure $(π<em>V)</em>*μ$ has a density $f_V$ satisfying $\int_{{f_V&gt;M}}f_V\,d\mathcal{L}<em>V^k\lesssim e^{-c(\log M)<sup>{1/3}}$ as $M\to\infty$, uniformly in $V$. Under strong separation, the conditional measures on the fibres are almost surely exact dimensional of dimension $\dim</em>{\mathrm{H}}μ-k$. The key novelty is to apply Varjú's $L<sup>2$ estimate under dense rotations to $L<sup>1$ smoothing increments of martingale-difference type similarly as Fourier decay is studied. This method requires no uniform spectral gap assumption.