Positive measure for restricted projections along smooth non-degenerate curves

Abstract

Let $1\leq m<n$, and let $γ:[0,1]\to\mathbb R^n$ be a smooth non-degenerate curve. For $θ\in[0,1]$, set \[ V_θ := \operatorname{span} \{γ&#39;(θ),\ldots,γ^{(m)}(θ)\}, \] and let $π_θ$ denote the orthogonal projection onto $V_θ$. 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 $θ\in[0,1]$.