Optimal quality of exceptional points for the Lebesgue density theorem

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.