  We consider certain finite sets of circle-valued functions defined on
intervals of real numbers and estimate how large the intervals must be for the
values of these functions to be uniformly distributed in an approximate way.
  This is used to establish some general conditions under which a random
construction introduced by Katznelson for the integers yields sets that are
dense in the Bohr group. We obtain in this way very sparse sets of real numbers
(and of integers) on which two different almost periodic functions cannot
agree, what makes them amenable to be used in sampling theorems for these
functions. These sets can be made as sparse as to have zero asymptotic density
or as to be t-sets, i.e., to be sets that intersect any of their translates in
a bounded set. Many of these results are proved not only for almost periodic
functions but also for classes of functions generated by more general complex
exponential functions, including chirps.
