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On the speed of convergence in the strong density theorem. (English) Zbl 1426.26004
Real Anal. Exch. 44, No. 1, 167-180 (2019); corrigendum ibid. 45, No. 2, 487-488 (2020).
This work constitutes a contribution to Problem 146 of Ulam and Erdős’s Scottish Book problems [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.

MSC:
26A12 Rate of growth of functions, orders of infinity, slowly varying functions
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
40A05 Convergence and divergence of series and sequences
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