  Measure rigidity is a branch of ergodic theory that has recently contributed
to the solution of some fundamental problems in number theory and mathematical
physics. Examples are proofs of quantitative versions of the Oppenheim
conjecture, related questions on the spacings between the values of quadratic
forms, a proof of quantum unique ergodicity for certain classes of hyperbolic
surfaces, and an approach to the Littlewood conjecture on the nonexistence of
multiplicatively badly approximable numbers.
  In these lectures we discuss a few simple applications of one of the central
results in measure rigidity: Ratner's theorem. We shall investigate the
statistical properties of certain number theoretic sequences, specifically the
fractional parts of $m\alpha$, $m=1,2,3,...$, (a classical, well understood
problem) and of $\sqrt{m\alpha}$ (as recently studied by Elkies and McMullen).
By exploiting equidistribution results on a certain homogeneous space, we will
show that the statistical properties of these sequences can exhibit significant
deviations from those of independent random variables. The ``randomness'' of
other, more generic sequences such as $m^2\alpha$ and $2^m \alpha$ mod 1 has
been studied extensively.
  These notes are based on lectures presented at the Institute Henri Poincare
Paris, June 2005, and at the summer school `Equidistribution in number theory',
CRM Montreal, July 2005. The author gratefully acknowledges support by an EPSRC
Advanced Research Fellowship.
