Positive measure for restricted projections along smooth non-degenerate curves

Abstract

Let $1\leq m<n$, and let $\gamma:[0,1]\to\mathbb R^n$ be a smooth non-degenerate curve. For $\theta\in[0,1]$, set \[ V_\theta := \operatorname{span} \{\gamma'(\theta),\ldots,\gamma^{(m)}(\theta)\}, \] 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 \[ \mathcal H^m(\pi_\theta(E))>0 \] for almost every $\theta\in[0,1]$.