On the speed of convergence in the strong density theorem

TL;DR

This paper establishes an elementary estimate on the convergence speed of the strong density theorem for compact sets in Euclidean space, refining understanding of how quickly density ratios approach one almost everywhere.

Contribution

It provides a new quantitative estimate on the rate of convergence in the strong density theorem, extending previous work and addressing a classical problem in measure theory.

Findings

Convergence rate is greater than 1 minus a small term involving the logarithm of the interval diameter.

The estimate holds for almost every point in the set as the interval diameter tends to zero.

The work relates to longstanding problems posed by Ulam and Erdős in measure theory.

Abstract

For a compact set $K\subset \mathbb{R}^m$, we have two indexes given under simple parameters of the set $K$ (these parameters go back to Besicovitch and Taylor in the late 50'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 \[ \frac{|R\cap K|}{|R|}>1-o\bigg(\frac{1}{|\log d(R)|}\bigg) \qquad \text{for a.e.} \ \ x\in K \ \ \text{and for} \ \ d(R)\to 0 \] (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 Erd\"{o}s' 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.