18,402 research outputs found

    Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces

    Full text link
    The article contains a survey of results on length-commensurable and isospectral locally symmetric spaces and related problems in the theory of semi-simple algebraic groups.Comment: New material has been added in section

    Best possible rates of distribution of dense lattice orbits in homogeneous spaces

    Get PDF
    The present paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup Γ\Gamma in a connected Lie (or algebraic) group GG, acting on suitable homogeneous spaces G/HG/H. The upper bound is derived using a quantitative duality principle for homogeneous spaces, reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on HH and acting on G/ΓG/\Gamma. In particular, the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of HH in the automorphic representation on L2(Γ∖G)L^2(\Gamma\setminus G). We show that the rate is best possible when the representation in question is tempered, and show that the latter condition holds in a wide range of examples

    Reducible quasi-periodic solutions for the Non Linear Schr\"odinger equation

    Full text link
    The present paper is devoted to the construction of small reducible quasi--periodic solutions for the completely resonant NLS equations on a dd--dimensional torus \T^d. The main point is to prove that prove that the normal form is reducible, block diagonal and satisfies the second Melnikov condition block wise. From this we deduce the result by a KAM algorithm.Comment: 48 page
    • …
    corecore