Optimal quality of exceptional points for the Lebesgue density theorem

TL;DR

This paper determines the maximum possible value of a density parameter related to the Lebesgue density theorem, resolving a problem previously studied by Kolyada and others.

Contribution

It provides a definitive solution to the problem of finding the optimal density value in the Lebesgue density theorem context.

Findings

Identifies the supremum of the density parameter δ.

Establishes the exact optimal value of δ.

Advances understanding of density points in measure theory.

Abstract

In spite of the Lebesgue density theorem, there is a positive $\delta$ such that, for every non-trivial measurable set $S$ of real numbers, there is a point at which both the lower densities of $S$ and of the complement of $S$ are at least $\delta$. The problem of determining the supremum of possible values of this $\delta$ was studied in a paper of V. I. Kolyada, as well as in some recent papers. We solve this problem in the present work.