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

TL;DR

This paper proves a density estimate for measurable sets in Euclidean space related to a conjecture by Zygmund, linking measure-theoretic properties to Lipschitz mappings and geometric structure.

Contribution

It establishes a lower bound on the density of a set along Lipschitz images, confirming a conjecture of Zygmund on Lipschitz differentiation and measure-zero characterizations.

Findings

Density estimate holds for almost every point in the set.

Set is negligible iff its intersection with Lipschitz images has measure zero.

Supports Zygmund's conjecture 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 $x \in A$.