2019 On the Speed of Convergence in the Strong Density Theorem
Panagiotis Georgopoulos, Constantinos Gryllakis
Real Anal. Exchange 44(1): 167-180 (2019). DOI: 10.14321/realanalexch.44.1.0167


For a compact set \(K\subseteq\mathbb{R}^m\), we have two indexes given under simple parameters of the set \(K\) (these parameters go back to Besicovitch and Taylor in the late 1950’s). In the present paper we prove that with the exception of a single extreme value for each index, we have the following elementary estimate on how fast the ratio in the strong density theorem of Saks will tend to one: \[ \frac{|R\cap K|}{|R|}>1-o\bigg(\frac{1}{|\log d(R)|}\bigg) \qquad \text{for a.e.} \ \ x\in K \ \ \text{and for} \ \ d(R)\to0 \] (provided \(x\in R\), where \(R\) is an interval in \(\mathbb{R}^m\), \(d\) stands for the diameter, and \(|\cdot|\) is the Lebesgue measure).

This work is a natural sequence of [3] and constitutes a contribution to Problem 146 of Ulam [5, p. 245] (see also [8, p.78]) and Erdös' Scottish Book ‘Problems’ [5, Chapter 4, pp. 27-33], since it is known that no general statement can be made on how fast the density will tend to one.


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Panagiotis Georgopoulos. Constantinos Gryllakis. "On the Speed of Convergence in the Strong Density Theorem." Real Anal. Exchange 44 (1) 167 - 180, 2019. https://doi.org/10.14321/realanalexch.44.1.0167


Published: 2019
First available in Project Euclid: 27 June 2019

zbMATH: 07088969
MathSciNet: MR3951340
Digital Object Identifier: 10.14321/realanalexch.44.1.0167

Primary: 26A12 , 28A05
Secondary: 40A05

Keywords: Besicovitch-Taylor index , Saks' strong density theorem , Speed of convergence

Rights: Copyright © 2019 Michigan State University Press


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Vol.44 • No. 1 • 2019
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