KTH/SU Mathematics Colloquium
06-09-13
Jens Marklof, University of Bristol
Distribution modulo one and ergodic theory
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 this introductory
lecture 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 $m\alpha$, $m=1,2,3,\ldots$, (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.