A lower bound on the quantitative version of the transversality theorem

TL;DR

This paper establishes a lower bound for the measure of the set where a continuous function intersects a manifold, providing a quantitative refinement of the transversality theorem.

Contribution

It introduces a sharp lower bound on the Hausdorff measure of the intersection set, advancing the quantitative understanding of transversality.

Findings

Proves a lower bound on the Hausdorff measure of the intersection set.

Provides a quantitative estimate in terms of power functions.

Enhances the theoretical understanding of transversality in analysis.

Abstract

The present paper studies a quantitative version of the transversality theorem. More precisely, given a continuous function $f\in \mathcal{C}([0,1]^d,\mathbb{R}^m)$ and a manifold $W\subset \mathbb{R}^m$ of dimension $p$, a sharpness result on the upper quantitative estimate of the $(d+p-m)$-dimensional Hausdorff measure of the set $\mathcal{Z}_{W}^{f}=\left\{x\in [0,1]^d: f(x)\in W\right\}$, which was achieved in [8], will be proved in terms of power functions.