## Real Analysis Exchange

### On the Speed of Convergence in the Strong Density Theorem

#### 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

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

#### 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