On metric Diophantine approximation in matrices and Lie groups

Abstract

Abstract. We study the diophantine exponent of analytic submanifolds of m × n real matrices, answering questions of Beresnevich, Kleinbock and Margulis. We identify a family of algebraic obstructions to the extremality of such a submanifold, and give a formula for the exponent when the submanifold is algebraic and defined over Q. We then apply these results to the determination of the diophantine exponent of rational nilpotent Lie groups.

On Diophantine transference principles

“…The present theory corresponds to the special case of n × 1 matrices. We refer the reader to [13,2,1,6] for recent developments on this theme. One of the main difficulties in studying Diophantine approximation on submanifolds of matrices is that it doesn't seem straightforward to define the correct notion of nondegeneracy for submanifolds or indeed the right generalization of friendly measures.…”

Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures

“…We also prove the result stated above that every weakly quasidecaying measure is strongly extremal (in particular verifying a conjecture of KLW), as well as considering the approximation properties of quasi-decaying measures on the space of matrices and on affine subspaces of R d . In particular we generalize results of some recent papers regarding approximation of friendly measures in the matrix framework [1,15] (cf. Theorems 1.7 and 1.9 below).…”

“…But any t ∈ a which shrinks u A r 1 to a small size also shrinks u A r 2 to a small size. In [15] this problem was circumvented by finding another approximant which can be shrunk to small size using only t ∈ a + , but this approximant may not be of as good quality as r 1 . In some sense the real problem might be that the function appearing in the definitions of ω and ω × does not give enough information as to how far a lattice is into the cusp.…”

“…This class includes a subclass of quasi-decaying measures, which are the analogue of KLW's "absolutely friendly" measures. 1 In the current paper (Part I), we demonstrate the most basic properties of the quasidecay condition, including the facts that every exact dimensional measure is quasidecaying, and that every quasi-decaying measure is extremal, which we prove using an elementary argument. We also prove the result stated above that every weakly quasidecaying measure is strongly extremal (in particular verifying a conjecture of KLW), as well as considering the approximation properties of quasi-decaying measures on the space of matrices and on affine subspaces of R d .…”

Equidistribution of expanding curves in homogeneous spaces and Diophantine approximation on square matrices

“…If ϕ satisfies condition A.1 above, then for every s 0 ∈ I such that ϕ (1) (s 0 ) is invertible, there always exists some subinterval J s0 of I, such that ϕ(s) − ϕ(s 0 ) is invertible for all s ∈ J s0 . This is because when s is close to s 0 , the major part of ϕ(s) − ϕ(s 0 ) is (s − s 0 )ϕ (1) (s 0 ) which is invertible. Thus the essential part of condition A.2 is that {(ϕ(s) − ϕ(s 0 )) −1 : s ∈ J} is not contained in a proper affine subspace of M(n × n, R).…”