18,402 research outputs found
Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces
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
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 in a
connected Lie (or algebraic) group , acting on suitable homogeneous spaces
. 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 and acting on
. In particular, the quality of the upper bound on the rate of
distribution we obtain is determined explicitly by the spectrum of in the
automorphic representation on . 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
The present paper is devoted to the construction of small reducible
quasi--periodic solutions for the completely resonant NLS equations on a
--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
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