Positive measure for restricted projections along smooth non-degenerate curves

Abstract

Let $1\le m<n$, and let $\gamma :[0,1]\to {\mathbb{R}}^n$ be a smooth non-degenerate curve. For $\theta \in [0,1]$, set \[ V_θ := \operatorname{span} \{γ'(θ),\ldots,γ^{(m)}(θ)\}, \] and let ${\pi }_{\theta }$ denote the orthogonal projection onto $V_{\theta }$. We prove that if $E\subset {\mathbb{R}}^n$ is a Borel set with ${\dim }_{\mathrm{H}}E>m$, then \[ \mathcal H^m(π_θ(E))>0 \] for almost every $\theta \in [0,1]$.

26 pages, corollary 1.3 is added