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Distribution of shapes of orthogonal lattices

Published online by Cambridge University Press:  02 November 2017

MANFRED EINSIEDLER
Affiliation:
D-Math, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland email manfred.einsiedler@math.ethz.ch, philipp.wirth@math.ethz.ch
RENÉ RÜHR
Affiliation:
Department of Mathematics, University of Tel Aviv, 69978 Tel-Aviv, Israel email reneruehr@gmail.com
PHILIPP WIRTH
Affiliation:
D-Math, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland email manfred.einsiedler@math.ethz.ch, philipp.wirth@math.ethz.ch

Abstract

It was recently shown by Aka, Einsiedler and Shapira that if $d>2$, the set of primitive vectors on large spheres when projected to the $(d-1)$-dimensional sphere coupled with the shape of the lattice in their orthogonal complement equidistribute in the product space of the sphere with the space of shapes of $(d-1)$-dimensional lattices. Specifically, for $d=3,4,5$ some congruence conditions are assumed. By using recent advances in the theory of unipotent flows, we effectivize the dynamical proof to remove those conditions for $d=4,5$. It also follows that equidistribution takes place with a polynomial error term with respect to the length of the primitive points.

Type
Original Article
Copyright
© Cambridge University Press, 2017 

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