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

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