We show that the set of oscillatory motions in the Sitnikov restricted three body problem has full Hausdorff dimension. This is a joint work with V.Kaloshin.
A classical theory of Dvoretsky states that every
infinite dimensional Banach space contains subspaces of arbitrarily large finite dimension,
which are arbitrarily close to a Hilbert space (in terms of the Banach-Mazur distance). V.Milman's proof of this result, based on the "concentration of measure" phenomenon, will be presented.