  This paper provides a rigorous derivation of a counterexample of Bourgain,
related to a well-known question of pointwise a.e. convergence for the solution
of the linear Schr\"odinger equation, for initial data in a Sobolev space. This
counterexample combines ideas from analysis and number theory, and the present
paper demonstrates how to build such counterexamples from first principles, and
then optimize them.
