Remarks on the box problem (Q1864717)

In connection with the study of Fourier integral operators there arises the question for an upper bound of the measure of the set \[ S= \{x\in Q:|f(x)|< 1\}, \] where \(Q\) is the unit cube and \(f\) is a \(C^n\) function on \(Q\) satisfying \(\partial^n_{x_1x_2\cdots x_n}f>\Lambda\). Related to this question the authors define a (CCW)-condition to given sets. They have an upper bound and a lower bound for \(|S|\) in the form \(O(\Lambda^{2^{1-n}})\) and \(O(\Lambda^{n^{-1}})\). Finally, they study a discrete analogue of the above question. This question was stated by Erdős in the context of hypergraphs.