A lower bound on the quantitative version of the transversality theorem

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 ${\mathcal{Z}}_W^f=\{x\in [0,1{]}^d:f(x)\in W\}$, which was achieved in [8], will be proved in terms of power functions.