An inverse theorem on 'economic' maps (Q2901897)

The authors deal with `economic' maps to Euclidean spaces; the quotes are apt as the notion is left mostly undefined. The reader can get an idea of what such a map might be by considering the main results of this paper; these are of the form: if \(\dim X\leq n\) then for most maps from~\(X\) to~\(\mathbb{R}^m\) and certain numbers~\(d\) the preimages of \(d\)-dimensional hyperplanes are small in some sense. Various theorems of this kind can be found in [\textit{S. A. Bogatyi} and \textit{V. M. Valov}, Sb. Math. 196, No. 11, 1585--1603 (2005); translation from Mat. Sb. 196, No. 11, 33--52 (2005; Zbl 1141.54014)]. In this paper the authors prove that certain bounds in the latter paper are sharp.