Regularity of almost-surely injective projections in Euclidean spaces

It is known that if a finite Borel measure $\mu$ in a Euclidean space has Hausdorff dimension smaller than a positive integer $k$, then the orthogonal projection onto almost every $k$-dimensional linear subspace is injective on a set of full $\mu$-measure. We study the regularity of the inverses of such projections. We prove that if $\mu$ has a compact support $X$ and (respectively) the Hausdorff, upper box-counting or Assouad dimension of $X$ is smaller than $k$, then the inverse is (respectively) continuous, pointwise H\"older for some $\alpha \in (0,1)$ or pointwise H\"older for every $\alpha \in (0,1)$. The result generalizes to the case of typical linear perturbations of Lipschitz maps. Additionally, we construct a non-trivial measure on the plane which admits almost-surely injective projections in every direction, and show that no homogeneous self-similar measure has this property.