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The present paper studies a quantitative version of the transversalitytheorem. More precisely, given a continuous function $f\in\mathcal{C}([0,1]^d,\mathbb{R}^m)$ and a manifold $W\subset \mathbb{R}^m$ ofdimension $p$, 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 wasachieved in [8], will be proved in terms of power functions.

A lower bound on the quantitative version of the transversality theorem