On a Lebesgue density property of subsets of real numbers related to Smítal's lemma (Q1910057)

The main result: Let \(A\) and \(B\) be sets of real numbers with positive outer Lebesgue measure, and let \(a\) and \(b\) be their respective outer density points. Let \(f\) be a real-valued function of two variables, which is locally of class \({\mathcal C}^1\) at \((a,b)\), and has nonzero partial derivatives at \((a,b)\). Then there is a nonempty open interval \(I\) such that the outer Lebesgue measure of \(f(A\times B)\cap J\) equals the diameter of \(J\), for any subinterval \(J\) of \(I\). This theorem generalizes similar results, proved in the case when, e.g., \(f(x,y)=x+y\) [cf. \textit{H. I. Miller}, J. Math. Anal. Appl. 124, 27-32 (1987; Zbl 0638.28001)].