Dimension estimates for the set of points with non-dense orbit in homogeneous spaces

Abstract

Let X = G/Γ, where G is a Lie group and Γ is a lattice in G, and let U be a subset of X whose complement is compact. We use the exponential mixing results for diagonalizable flows on X to give upper estimates for the Hausdorff dimension of the set of points whose trajectories miss U . This extends a recent result of Kadyrov [10] and produces new applications to Diophantine approximation, such as an upper bound for the Hausdorff dimension of the set of weighted uniformly badly approximable systems of linear forms, generalizing an estimate due to Broderick and Kleinbock [2].

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

“…The following theorem is an immediate corollary of [KMi,Theorem 4.1] applied to P = H, L = dim P = mn and U = S c . Theorem 3.1.…”

“…In this section we will consider the case of S being compact, which was thoroughly studied in [KMi]. We are going to apply [KMi,Theorem 4.1], which was proved in the generality of X = G/Γ being an arbitrary homogeneous space, and H being a subgroup of G with the Effective Equidistribution Property (EEP) with respect to F + . The latter property was shown there to hold in the case (1.2)-(1.3), or, more generally, as long as H is the expanding horospherical subgroup relative to F + , and the F + -action on X is exponentially mixing.…”

“…where C 4 = 1 C 8 α (m+n−1) 2(m+n−1) . Recall that 1 α 1 (x) is equal to the norm of the shortest vector in the lattice x; therefore by [KMi,Lemma 7.2], r 0 X t ≤C 4 α e mnt is at least…”

Dimension drop for diagonalizable flows on homogeneous spaces

“…it is reasonable to conjecture that the answer is always affirmative; in other words, that the following 'Dimension Drop Conjecture' (DDC) holds: if F ⊂ G is a subsemigroup and O is an open subset of X, then either E(F, O) has positive measure, or its dimension is less than the dimension of X. When X is compact it follows from the variational principle for measure-theoretic entropy, as outlined in [KW2,§7]; an effective argument using exponential mixing was developed in [KMi1].…”

Dimension estimates for badly approximable affine forms

“…The sets Bad b and Bad A are unions of subsets Bad b pǫq and Bad A pǫq over ǫ ą 0, respectively, thus a more refined question is whether the Hausdorff dimension of Bad b pǫq, Bad A pǫq could still be of full dimension. For the homogeneous case (b " 0), the Hausdorff dimension Bad 0 pǫq is less than the full dimension mn (see [BK13,Sim18] for the unweighted case and [KM19] for the weighted case). Thus, a natural question is whether Bad b pǫq can have full Hausdorff dimension for some b.…”