Exceptional points for Lebesgue's density theorem on the real line

For a nontrivial measurable set on the real line, there are always exceptional points, where the lower and upper densities of the set are neither 0 nor 1. We quantify this statement, following work by V. Kolyada, and obtain the unexpected result that there is always a point where the upper and the lower densities are closer to 1/2 than to zero or one. The method of proof uses a discretized restatement of the problem, and a self-similar construction.