Regularity of almost-surely injective projections in Euclidean spaces

TL;DR

This paper investigates the regularity of inverse projections of measures in Euclidean spaces, showing conditions under which these inverses are continuous or Hölder continuous, and providing examples and counterexamples to illustrate the results.

Contribution

It extends previous work by analyzing the regularity of inverse projections under various dimensional constraints and generalizes to typical linear perturbations of Lipschitz maps.

Findings

Inverse projections are continuous if the measure's support has Hausdorff, box-counting, or Assouad dimension less than k.

Inverse projections are pointwise Hölder continuous with some exponent α under certain dimensional conditions.

Constructs a measure with almost-surely injective projections in all directions, unlike homogeneous self-similar measures.

Abstract

In a previous work we proved 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. In this paper we study the regularity of the inverses of these projections and prove that if $\mu$ has a compact support $X$ such that (respectively) the Hausdorff, upper box-counting or Assouad dimension of $X$ is smaller than $k$, then the inverse is (respectively) continuous, pointwise $\alpha$-H\"older for some $\alpha \in (0,1)$ or pointwise $\alpha$-H\"older for every $\alpha \in (0,1)$. The results generalize to the case of typical linear perturbations of Lipschitz maps and strengthen previously known ones in the lossless analog compression literature. We provide examples showing the sharpness of the statements. 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.