Optimal quality of exceptional points for the Lebesgue density theorem

Abstract

In spite of the Lebesgue density theorem, there is a positive d 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 d. The problem of determining the supremum of possible values of this d was studied in a paper of V.

I. Kolyada, as well as in some recent papers.

We solve this problem in the present work.

Keywords

• Optimal
• quality
• exceptional
• points
• for
• Lebesgue
• density
• theorem