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Homogeneous orbit closures and applications

Published online by Cambridge University Press:  28 April 2011

ELON LINDENSTRAUSS and
URI SHAPIRA
ELON LINDENSTRAUSS
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel (email: elon@math.huji.ac.il)
URI SHAPIRA
Affiliation:
ETH Zürich, Departement Mathematik, Rämistrasse 101, 8092 Zürich, Switzerland (email: ushapira@gmail.com)
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Abstract

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We give new classes of examples of orbits of the diagonal group in the space of unit volume lattices in ℝd for d≥3 with nice (homogeneous) orbit closures, as well as examples of orbits with explicitly computable but irregular orbit closures. We give Diophantine applications to the former; for instance, we show that, for all γ,δ∈ℝ, \[ \liminf _{|n|\to \infty }|n|\langle n \sqrt [3]{2}-\gamma \rangle \langle n \sqrt [3]{4}-\delta \rangle =0, \] where 〈c〉 denotes the distance of a real number c to the integers.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012, pp. 785 - 807
Copyright
Copyright © Cambridge University Press 2011

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