Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Asymptotic Problems on Homogeneous Spaces
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This PhD thesis consists of a summary and five papers which all deal with asymptotic problems on certain homogeneous spaces.

In Paper I we prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. All our results are given with precise estimates on the rates of convergence to equidistribution.

Papers II and III are concerned with statistical problems on the space of n-dimensional lattices of covolume one. In Paper II we study the distribution of lengths of non-zero lattice vectors in a random lattice of large dimension. We prove that these lengths, when properly normalized, determine a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity 1/2. In Paper III we complement this result by proving that the asymptotic distribution of the angles between the shortest non-zero vectors in a random lattice is that of a family of independent Gaussians.

In Papers IV and V we investigate the value distribution of the Epstein zeta function along the real axis. In Paper IV we determine the asymptotic value distribution and moments of the Epstein zeta function to the right of the critical strip as the dimension of the underlying space of lattices tends to infinity. In Paper V we determine the asymptotic value distribution of the Epstein zeta function also in the critical strip. As a special case we deduce a result on the asymptotic value distribution of the height function for flat tori. Furthermore, applying our results we discuss a question posed by Sarnak and Strömbergsson as to whether there in large dimensions exist lattices for which the Epstein zeta function has no zeros on the positive real line.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics , 2010. , p. 24
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 68
Keywords [en]
Analysis on homogeneous spaces, hyperbolic manifolds, spectral theory, equidistribution, the space of lattices, length statistics, Poisson process, moments, Epstein zeta function, value distribution, height function.
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-132645ISBN: 978-91-506-2163-1 (print)OAI: oai:DiVA.org:uu-132645DiVA, id: diva2:359184
Public defence
2010-12-09, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Funder
Available from: 2010-11-17 Created: 2010-10-23 Last updated: 2010-11-17Bibliographically approved
List of papers
1. On the uniform equidistribution of closed horospheres in hyperbolic manifolds
Open this publication in new window or tab >>On the uniform equidistribution of closed horospheres in hyperbolic manifolds
2012 (English)In: Proceedings of the London Mathematical Society, ISSN 0024-6115, E-ISSN 1460-244X, Vol. 105, p. 225-280Article in journal (Refereed) Published
Abstract [en]

We prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. This extends earlier results by Hejhal ['On the uniform equidistribution of long closed horocycles', Loo-Keng Hua: a great mathematician of the twentieth century, Asian J. Math. 4 (2000) 839-853] and Strombergsson ['On the uniform equidistribution of long closed horocycles', Duke Math. J. 123 (2004) 507-547] in dimension 2. Our proofs use spectral methods, and lead to precise estimates on the rate of convergence to equidistribution.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-132643 (URN)10.1112/plms/pdr052 (DOI)000307174700001 ()
Available from: 2010-10-23 Created: 2010-10-23 Last updated: 2017-12-12Bibliographically approved
2. On the Poisson distribution of lengths of lattice vectors in a random lattice
Open this publication in new window or tab >>On the Poisson distribution of lengths of lattice vectors in a random lattice
2011 (English)In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 269, no 3-4, p. 945-954Article in journal (Refereed) Published
Abstract [en]

We prove that the volumes determined by the lengths of the non-zero vectors +/- x in a random lattice L of covolume 1 define a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity 1/2. This generalizes earlier results by Rogers (Proc Lond Math Soc (3) 6:305-320, 1956, Thm. 3) and Schmidt (Acta Math 102: 159-224, 1959, Satz 10).

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-132640 (URN)10.1007/s00209-010-0772-8 (DOI)000297355500019 ()
Available from: 2010-10-23 Created: 2010-10-23 Last updated: 2017-12-12Bibliographically approved
3. On the distribution of angles between the N shortest vectors in a random lattice
Open this publication in new window or tab >>On the distribution of angles between the N shortest vectors in a random lattice
2011 (English)In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 84, no 3, p. 749-764Article in journal (Refereed) Published
Abstract [en]

We determine the joint distribution of the lengths of, and angles between, the N shortest lattice vectors in a random n-dimensional lattice as n→∞. Moreover, we interpret the result in terms of eigenvalues and eigenfunctions of the Laplacian on flat tori. Finally, we discuss the limit distribution of any finite number of successive minima of a random n-dimensional lattice as n→∞.

National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-132641 (URN)10.1112/jlms/jdr032 (DOI)000297572800012 ()
Available from: 2010-10-23 Created: 2010-10-23 Last updated: 2017-12-12Bibliographically approved
4. On the value distribution and moments of the Epstein zeta function to the right of the critical strip
Open this publication in new window or tab >>On the value distribution and moments of the Epstein zeta function to the right of the critical strip
2011 (English)In: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 131, no 7, p. 1176-1208Article in journal (Refereed) Published
Abstract [en]

We study the Epstein zeta function E-n(L, S) for s > n/2 and a random lattice L of large dimension n. For any fixed c > 1/2 we determine the value distribution and moments of E-n(., cn) (suitably normalized) as n -> infinity. We further discuss the random function C -> E-n(., cn) for c is an element of [A, B] with 1/2 < A < B and determine its limit distribution as n -> infinity.

Keywords
Lattices, Epstein's zeta function, Value distribution, Moments
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-132642 (URN)10.1016/j.jnt.2010.12.003 (DOI)000289178500002 ()
Available from: 2010-10-23 Created: 2010-10-23 Last updated: 2017-12-12Bibliographically approved
5.
The record could not be found. The reason may be that the record is no longer available or you may have typed in a wrong id in the address field.

Open Access in DiVA

fulltext(1215 kB)383 downloads
File information
File name FULLTEXT01.pdfFile size 1215 kBChecksum SHA-512
ced94c681f93ba828e15c486037ad5fd3e155816103580115432ac5b9e321dd5a9d857c78e5965ba5add4fbb6269bc40379eb20736751b58b6c3e8306cac38db
Type fulltextMimetype application/pdf

By organisation
Department of Mathematics
Mathematics

Search outside of DiVA

GoogleGoogle Scholar
Total: 383 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 813 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf