Optimal quality of exceptional points for the Lebesgue density theorem

In spite of the Lebesgue density theorem, there is a positive $δ$ 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 $δ$. The problem of determining the supremum of possible values of this $δ$ was studied in a paper of V. I. Kolyada, as well as in some recent papers. We solve this problem in the present work.

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