Density estimate from below in relation to a conjecture of A. Zygmund on Lipschitz differentiation

Abstract

Letting $A \subset \mathbb{R}^n$ be Borel measurable and $W_0 : A \to \mathbb{G}(n,m)$ Lipschitzian, we establish that \begin{equation*} \limsup_{r \to 0^+} \frac{\mathcal{H}^m \left[ A \cap B(x,r) \cap (x+ W_0(x))\right]}{\alpha(m)r^m} \geq \frac{1}{2^n}, \end{equation*} for $\mathcal{L}^n$-almost every $x \in A$. In particular, it follows that $A$ is $\mathcal{L}^n$-negligible if and only if $\mathcal{H}^m(A \cap (x+W_0(x))=0$, for $\mathcal{L}^n$-almost every $x \in A$.