On the speed of convergence in the strong density theorem (Q2316562)

This work constitutes a contribution to Problem 146 of Ulam and Erdős's Scottish Book problems [\textit{R. D. Mauldin}, The Scottish Book. Mathematics from the Scottish Café. With selected problems from the New Scottish Book. 2nd updated and enlarged edition. Cham: Birkhäuser/Springer (2015; Zbl 1331.01039)], on how fast the ratio in the strong density theorem of Saks will tend to one. Under some technical conditions, one has: \[ \frac{|R \cap K|}{|R|} > 1- o \biggl( \frac{1}{|\log d(R)|}\biggr) \text{ for a.e. } x\in K \text{ and as } d(R) \rightarrow 0, \] where \(K\) is a compact set in \(\mathbb{R}^n\), \(R\) is an interval in \(\mathbb{R}^n\), \(d\) stands for the diameter, and \(\left| \cdot \right|\) is the Lebesgue measure.