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Title of Journal: Math Ann

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Abbravation: Mathematische Annalen

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Springer Berlin Heidelberg

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DOI

10.1002/chin.198815096

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1432-1807

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Pointwise equidistribution with an error rate and

Authors: Dmitry Kleinbock Ronggang Shi Barak Weiss
Publish Date: 2016/04/04
Volume: 367, Issue: 1-2, Pages: 857-879
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Abstract

Consider G=mathrmSL d mathbb R and Gamma =mathrmSL d mathbb Z It was recently shown by the secondnamed author Shi Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space preprint arXiv14052067 2014 that for some diagonal subgroups g tsubset G and unipotent subgroups Usubset G g ttrajectories of almost all points on all Uorbits on G/Gamma are equidistributed with respect to continuous compactly supported functions varphi on G/Gamma In this paper we strengthen this result in two directions by exhibiting an error rate of equidistribution when varphi is smooth and compactly supported and by proving equidistribution with respect to certain unbounded functions namely Siegel transforms of Riemann integrable functions on mathbb Rd For the first part we use a method based on effective double equidistribution of g ttranslates of Uorbits which generalizes the main result of Kleinbock and Margulis On effective equidistribution of expanding translates of certain orbits in the space of lattices Number theory analysis and geometry 385–396 2012 The second part is based on Schmidt’s results on counting of lattice points Numbertheoretic consequences involving spiraling of lattice approximations extending recent work of Athreya et al J Lond Math Soc 912383–404 2015 are derived using the equidistribution resultThe support of Grants NSFC 11201388 NSFC 11271278 ERC starter Grant DLGAPS 279893 NSF DMS1101320 NSF 0932078 000 and BSF 2010428 is gratefully acknowledged We would like to thank Manfred Einsiedler for a discussion of effective double equidistribution Marc Pollicott for pointing us to references 9 10 and MSRI for its hospitality during Spring 2015

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