Inverse images of open sets under gradient mapping

Abstract

For every pair of sets $${F, U \subset \mathbb{R}^d}$$ , $${d \geq 2}$$ , F being of Borel class Fσ and U being nonempty, bounded and open, we construct a Frechet differentiable function $${f \colon \mathbb{R}^{d} \to \mathbb{R}}$$ such that $${F \subset (\nabla{f})^{-1}(U)}$$ and the Hausdorff dimension of $${(\nabla{f})^{-1}(U) \setminus F}$$ does not exceed 1. Moreover $${(\nabla{f})(\mathbb{R}^d) \subset \overline{U}}$$ . This generalizes both Zelený [10] and Deville–Matheron [8] results about the properties of open sets preimages under the gradient mapping.