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

TL;DR

This paper proves that for a non-degenerate smooth curve in b^n, the orthogonal projection of a Borel set onto the tangent space at almost every point preserves the Hausdorff dimension up to the minimum of the set's dimension and the tangent space dimension.

Contribution

It establishes a dimension preservation result for projections of sets onto tangent spaces of non-degenerate smooth curves in b^n.

Findings

Projection preserves Hausdorff dimension for almost every point on the curve.

Dimension of projected set equals m or the original dimension, whichever is smaller.

Results extend classical projection theorems to tangent spaces of curves.

Abstract

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 $\theta\in [-1, 1]$.