90 research outputs found
Uniform distribution of orbits of lattices on spaces of frames
We study distribution of orbits of a lattice \Gamma<SL(n,R) in the the space
V_{n,l} of l-frames in R^n (1\le l\le n-1). Examples of dense \Gamma-orbits are
known from the work of Dani, Raghavan, and Veech. We show that dense orbits of
\Gamma are uniformly distributed in V_{n,l} with respect to an explicitly
described measure. We also establish analogous result for lattices in Sp(n,R)
that act on the space of isotropic n-frames.Comment: To be published in Duke Math. Journal, 31 page
Lattice action on the boundary of SL(n,R)
Let \Gamma be a lattice in G=SL(n,R) and X=G/S a homogeneous space of G,
where S is a closed subgroup of G which contains a real algebraic subgroup H
such that G/H is compact. We establish uniform distribution of orbits of \Gamma
in X analogous to the classical equidistribution on torus. To obtain this
result, we first prove an ergodic theorem along balls in the connected
component of Borel subgroup of G.Comment: 21 page
On Oppenheim-type conjecture for systems of quadratic forms
Let Q_i, i=1,...,t, be real nondegenerate indefinite quadratic forms in d
variables. We investigate under what conditions the closure of the set
{(Q_1(x),...,Q_t(x)): x\in Z^d-{0}} contains (0,..,0). As a corollary, we
deduce several results on the magnitude of the set \Delta of g\in GL(d,R) such
that the closure of the set {(Q_1(gx),...,Q_t(gx)): x\in Z^d-{0}} contains
(0,...,0). Special cases are described when depending on the mutual position of
the hypersurfaces {Q_i=0}, i=1,...,t, the set \Delta has full Haar measure or
measure zero and Hausdorff dimension d^2-(d-2)/2.Comment: To be published in Israel Journal of Mathematics, 19 page
Algebraic Numbers, Hyperbolicity, and Density Modulo One
We prove the density of the sets of the form modulo one,
where and are multiplicatively independent algebraic
numbers satisfying some additional assumptions. The proof is based on analysing
dynamics of higher-rank actions on compact abelean groups
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