uu.seUppsala University Publications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • 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
On the generalized circle problem for a random lattice in large dimension
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Univ Copenhagen, Dept Math Sci, Univ Pk 5, DK-2100 Copenhagen O, Denmark;Chalmers Univ Technol, Dept Math Sci, SE-41296 Gothenburg, Sweden;Univ Gothenburg, SE-41296 Gothenburg, Sweden.
2019 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 345, p. 1042-1074Article in journal (Refereed) Published
Abstract [en]

In this note we study the error term Rn,L(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary function f : Z+ → R+ satisfying limn→∞ f (n) = ∞ and f (n) = Oε(eεn) for every ε > 0. Then, the random function

t |→ 1/root 2f (n) Rn,L (t f(n))

on the interval [0, 1] converges in distribution to one-dimensional Brownian motion as n → ∞. The proof goes via convergence of moments, and for the computations we develop a new version of Rogers' mean value formula from [18]. For the individual kth moment of the variable (2f (n))-1/2 Rn,L (f (n)) we prove convergence to the corresponding Gaussian moment more generally for functions f satisfying f (n) = O(ecn) for any fixed c ∈ (0, ck), where ck is a constant depending on k whose optimal value we determine.

Place, publisher, year, edition, pages
2019. Vol. 345, p. 1042-1074
Keywords [en]
The generalized circle problem, Random lattice, Rogers' mean value formula, Brownian motion
National Category
Probability Theory and Statistics Mathematical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-379328DOI: 10.1016/j.aim.2019.01.034ISI: 000459529500027OAI: oai:DiVA.org:uu-379328DiVA, id: diva2:1298365
Funder
Swedish Research Council, 623-2011-743Swedish Research Council, 621-2011-3629EU, FP7, Seventh Framework Programme, DFF-1325-00058Göran Gustafsson Foundation for Research in Natural Sciences and MedicineAvailable from: 2019-03-22 Created: 2019-03-22 Last updated: 2019-03-22Bibliographically approved

Open Access in DiVA

No full text in DiVA

Other links

Publisher's full text

Authority records BETA

Strömbergsson, Andreas

Search in DiVA

By author/editor
Strömbergsson, Andreas
By organisation
Department of Mathematics
In the same journal
Advances in Mathematics
Probability Theory and StatisticsMathematical Analysis

Search outside of DiVA

GoogleGoogle Scholar

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 6 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • 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