  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
zero nor one. 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 combinatorial restatement of the problem.
