Metric diophantine approximation on Lie groups

A natural generalization of classical diophantine approximation consists in taking products of elements in a Lie group and ask how fast they can approach the identity. For example, classically, a number x is Diophantine if and only if sums of N terms each equal to 1, -1, x or -x, cannot get closer to zero than an inverse power of N. Replacing x and 1 by two or more vectors in a vector space is the subject of simultaneous approximation, and diophantine approximation in matrices. In this talk I will consider the case when the elements are chosen from a more general Lie group and focus on nilpotent groups. It turns out that the computation of optimal exponents for nilpotent Lie groups can be reduced to that of certain algebraic subvarieties of matrices. We give a formula for the exponent of arbitrary submanifolds of matrices, showing it depends only on their algebraic closure, answering a question of Kleinbock and Margulis, and use it to express the exponent of nilpotent Lie groups in terms of various representation theoretic data. This is joint work with M. Aka, L. Rosenzweig and N. de Saxce

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