On the dimension drop conjecture for diagonal flows on the space of lattices

Abstract

Let X = G/Γ, where G is a Lie group and Γ is a lattice in G, let U be an open subset of X, and let {gt} be a one-parameter subgroup of G. Consider the set of points in X whose gt-orbit misses U ; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture has been proved when X is compact or when G is a simple Lie group of real rank 1. In this paper we prove this conjecture for the case G = SLm+n(R), Γ = SLm+n(Z) and gt = diag(e nt , . . . , e nt , e −mt , . . . , e −mt ), in fact providing an effective estimate for the codimension. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on SLm+n(R)/ SLm+n(Z). We also discuss an application to the problem of improving Dirichlet's theorem in simultaneous Diophantine approximation.

Dimension drop for diagonalizable flows on homogeneous spaces

“…However combining the non-escape of mass argument with an additional construction taking care of the compact part of the space is more involved. Previously this was done in the case when G is a simple Lie group of real rank 1 [EKP], and then, in the most recent work of the authors [KMi2], when X = SL m+n (R)/ SL m+n (Z) and F = diag(e nt , . .…”

“…In this paper we generalize the approach of [KMi2] by exhibiting two abstract assumptions sufficient for the validity of DDC. One takes care of the compact part of the space, while the other deals with the non-escape of mass.…”

Abundance of Dirichlet-improvable pairs with respect to arbitrary norms

“…The exceptional sets similar to that in the right hand side of (4.7) were recently investigated in [KM], where, in particular, the so-called Dimension Drop Conjecture was solved for the a s -action on X 3 . More precisely, since K η (r) has non-empty interior, in view of [KM,Theorem 1.2] the set in the right hand side of (4.7) has less than full Hausdorff dimension, which contradicts Theorem 1.1.…”

A measure estimate in geometry of numbers and improvements to Dirichlet's theorem

“…The problem of improving Dirichlet's theorem was initiated by Davenport and Schmidt [10,9] where they showed that the set DI m,n := 0<c<1 DI m,n (cψ 1 ) (1.3) of Dirichlet improvable matrices is of Lebesgue measure zero, while having full Hausdorff dimension mn. More recently, Kleinbock and Mirzadeh [22,Theorem 1.5] showed that for any fixed 0 < c < 1, the Hausdorff dimension of DI m,n (cψ 1 ) is strictly smaller than mn. There have also been extensive studies on the Hausdorff dimensions of the (even smaller) set of the singular matrices, Sing m,n := 0<c<1 DI m,n (cψ 1 ).…”

“…It seems that by using Theorem 1.3 (cf. also Theorem 5.1 below), together with a further analysis of the quantities appearing in[22, Theorem 1.2], it should be possible to sharpen the conclusion of[22, Theorem 1.5] into a bound of the form…”