A restricted projection problem for fractal sets in ${\mathbb{R}}^n$

Let $\gamma :[-1,1]\to {\mathbb{R}}^n$ be a smooth curve that is non-degenerate. Take $m\le n$ and a Borel set $E\subset [0,1{]}^n$. We prove that the orthogonal projection of $E$ to the $m$-th order tangent space of $\gamma$ at $\theta \in [-1,1]$ has Hausdorff dimension $\min \{m,\dim (E)\}$ for almost every $\theta \in [-1,1]$.