Number Theory Seminar

Speaker: Lior Rosenzweig (KTH Stockholm)

Title: Diophantine approximation in nilpotent Lie groups

Abstract: In classical Diophantine approximation one asks how well a real number x can be approximated by an irreducible fraction whose numerator and denominator are integers. This formulation can be extended to the non-abelian world of Lie groups: A finitely generated subgroup of a real Lie group G is said to be Diophantine if there is b>0 such that non-trivial elements in the word ball B(n) centered at the identity never approach the identity of G closer than 1/|B(n)|^b, and the Lie group G is said to be Diophantine if for every k>0, a random k-tuple in G generates a Diophantine subgroup. In this talk I will discuss the case of nilpotent subgroups. We will show a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most 5, or derived length at most 2, as well as rational nilpotent Lie groups are Diophantine. Time permitting we will also discuss further progress of this question involving the optimality of the exponent b for a given nilpotent Diophantine group G, related to the works of Kleinbock and Margulis. This is joint work with M. Aka, E. Breuillard and N. de Saxce.