Real Analysis Exchange

On the Speed of Convergence in the Strong Density Theorem

Panagiotis Georgopoulos and Constantinos Gryllakis

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Abstract

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.

Article information

Source
Real Anal. Exchange, Volume 44, Number 1 (2019), 167-180.

Dates
First available in Project Euclid: 27 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.rae/1561622438

Digital Object Identifier
doi:10.14321/realanalexch.44.1.0167

Mathematical Reviews number (MathSciNet)
MR3951340

Zentralblatt MATH identifier
07088969

Subjects
Primary: 26A12: Rate of growth of functions, orders of infinity, slowly varying functions [See also 26A48] 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]
Secondary: 40A05: Convergence and divergence of series and sequences

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

Citation

Georgopoulos, Panagiotis; Gryllakis, Constantinos. On the Speed of Convergence in the Strong Density Theorem. Real Anal. Exchange 44 (2019), no. 1, 167--180. doi:10.14321/realanalexch.44.1.0167. https://projecteuclid.org/euclid.rae/1561622438


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