Generalizing Cusick’s theorem on the closedness of the classical Lagrange spectrum for the approximation of real numbers by rational ones, we prove that various approximation spectra are closed, using penetration properties of the geodesic flow in cusp neighbourhoods in negatively curved manifolds and a result of Maucourant [Mau].
Let be a Dedekind domain with field of fractions and a finite group. We show that, if is a ring of -adic integers, then the Witt decomposition map between the Grothendieck-Witt group of bilinear -modules and the one of finite bilinear -modules is surjective. For number fields is also surjective, if is a nilpotent group of odd order, but there are counterexamples for groups of even order.
We propose two conjectures which imply the Collatz conjecture. We give a numerical evidence for the second conjecture.