We prove a number field analogue of W. M. Schmidt's conjecture on the intersection of weighted badly approximable vectors and use this to prove an instance of a conjecture of An, Guan and Kleinbock [4]. Namely, let G := SL2(R) × · · · × SL2(R) and Γ be a lattice in G. We show that the set of points on G/Γ whose forward orbits under a one parameter Ad-semisimple subsemigroup of G are bounded, form a hyperplane absolute winning set.
Consider the homogeneous dynamical system (G/Γ, g) induced by the left translation of g ∈ G on the homogeneous space G/Γ, where G is a connected semisimple Lie group and Γ ⊂ G a cocompact lattice. We study the properties of the system (G/Γ, g) related to almost weak specification. Based on the work of Quas and Soo, we prove Katok's conjecture on intermediate metric entropies for the system (G/Γ, g) for almost all g ∈ G (with respect to the Haar measure on G), i.e. the setMoreover, we show that for a topological dynamical system (Y, f ) which satisfies almost weak specification and asymptotically entropy expansiveness, the set E(Y, f ) defined as above is dense in [0, h top ( f )).
We extend the Khintchine transference inequalities, as well as a homogeneous-inhomogeneous transference inequality for lattices, due to Bugeaud and Laurent, to a weighted setting. We also provide applications to inhomogeneous Diophantine approximation on manifolds and to weighted badly approximable vectors. Finally, we interpret and prove a conjecture of Beresnevich-Velani (2010) about inhomogeneous intermediate exponents.
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