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The number of points from a random lattice that lie inside a ball
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot hold if one averages over the space of all lattices.

Keyword [en]
geometry of numbers, lattices
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-177287OAI: oai:DiVA.org:kth-177287DiVA: diva2:872108
Funder
Swedish Research Council
Note

QC 20160214

Available from: 2015-11-17 Created: 2015-11-17 Last updated: 2016-02-14Bibliographically approved
In thesis
1. Geometry of numbers, class group statistics and free path lengths
Open this publication in new window or tab >>Geometry of numbers, class group statistics and free path lengths
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis contains four papers, where the first two are in the area of geometry of numbers, the third is about class group statistics and the fourth is about free path lengths. A general theme throughout the thesis is lattice points and convex bodies.

In Paper A we give an asymptotic expression for the number of integer matrices with primitive row vectors and a given nonzero determinant, such that the Euclidean matrix norm is less than a given large number. We also investigate the density of matrices with primitive rows in the space of matrices with a given determinant, and determine its asymptotics for large determinants.

In Paper B we prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot hold if one averages over the space of all lattices.

In Paper C, we give a conjectural asymptotic formula for the number of imaginary quadratic fields with class number h, for any odd h, and a conjectural asymptotic formula for the number of imaginary quadratic fields with class group isomorphic to G, for any finite abelian p-group G where p is an odd prime. In support of our conjectures we have computed these quantities, assuming the generalized Riemann hypothesis and with the aid of a supercomputer, for all odd h up to a million and all abelian p-groups of order up to a million, thus producing a large list of “missing class groups.” The numerical evidence matches quite well with our conjectures.

In Paper D, we consider the distribution of free path lengths, or the distance between consecutive bounces of random particles in a rectangular box. If each particle travels a distance R, then, as R → ∞ the free path lengths coincides with the distribution of the length of the intersection of a random line with the box (for a natural ensemble of random lines) and we determine the mean value of the path lengths. Moreover, we give an explicit formula for the probability density function in dimension two and three. In dimension two we also consider a closely related model where each particle is allowed to bounce N times, as N → ∞, and give an explicit formula for its probability density function.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2015. vi, 13 p.
Series
TRITA-MAT-A, 2015:15
Keyword
geometry of numbers, lattices, class numbers, class groups, free path lengths
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-177888 (URN)978-91-7595-797-5 (ISBN)
Public defence
2016-01-15, F3, Lindstedsvägen 26, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council
Note

QC 20151204

Available from: 2015-12-04 Created: 2015-11-30 Last updated: 2015-12-04Bibliographically approved

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Citation style
  • apa
  • harvard1
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  • vancouver
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