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

Abstract

Let $A\subset\mathbb{R}^d$, $d\ge3$, be a self-similar set whose defining rotations generate a dense subgroup of $\mathrm{SO}(d)$. For every integer $1\le k<\dim_{\mathrm{H}} A$, we prove that its orthogonal projections $π_V A$ have uniformly positive $k$-dimensional Lebesgue measure, and that their fibres have Hausdorff dimension $\dim_{\mathrm{H}} A-k$ 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 $t>k$. Every projected measure $(π_V)_*μ$ has a density $f_V$ satisfying $\int_{\{f_V>M\}}f_V\,d\mathcal{L}_V^k\lesssim e^{-c(\log M)^{1/3}}$ as $M\to\infty$, uniformly in $V$. Under strong separation, the conditional measures on the fibres are almost surely exact dimensional of dimension $\dim_{\mathrm{H}}μ-k$. The key novelty is to apply Varjú's $L^2$ estimate under dense rotations to $L^1$ smoothing increments of martingale-difference type similarly as Fourier decay is studied. This method requires no uniform spectral gap assumption.