Beginning with nothing more complicated than rotating some starting point x by some irrational rotation α, we can try to find those times n so that of the first n iterates of x under the rotation, exactly half have landed in the bottom half of the circle, and exactly half have landed in the top half.

Abstract results in ergodic theory tell us that the set of such n will generally be infinite but "sparse," in some sense. We can investigate the sparseness by trying to sum 1/n over this set, and seeing if we can expect to form a convergent or divergent series.