The lattice discrepancy of large domains
- Boden und Landökosysteme
- Wasser - Atmosphäre - Umwelt
- Lebensraum und Landschaft
- Lebensmittel, Ernährung, Gesundheit
- Ressourcen und gesellschaftliche Dynamik
- Forschungscluster "Nachhaltigkeit"
Abstract
Let B be a body in s-dimensional Euclidean space with smooth boundary throughout, and let t be a large real parameter by which the body B is dilated ("blown up"). The number of integer points in the enlarged body tB is approximated in first order by the volume of tB. The question for the error involved in this approximation – the so-called “lattice point discrepancy” - has created a deep and prolific mathematical theory. In the planar case (s = 2), the situation is comparatively well understood, as well as for bodies in three-dimensional space that are sort of “egg-shaped” (i.e., connvex with smooth boundary of nonzero curvature throghout, in mathematical terms). Therefore, the main objective of the project submitted is to focus on the most explicit deviation from convexity that a body can have, namely a "dent", the boundary being still smooth. It is expected that there will be obtained rather precise asymptotic formulas for the lattice discrepancy. These investigations are to be carried for in dimension 3 and bigger. The whole theme is a natural and central question of multivariate analysis. It can be viewed as sort of a basic science for multidimensional numerical integration, it has connections to Fourier theory, to the classic analytic theory of special arithmetic functions, to Diophantine equations, and to the theory of linear operators. Furthermore, there are even interactions, resp., applications to quantum physics and to crystallography.
seminariomatematico.dm.unito.it
Project staff
Werner Georg Nowak
em.o.Univ.Prof. Dr.phil. Werner Georg Nowak
Project Leader
01.02.2009 - 31.01.2012
Gerald Kuba
ao.Univ.Prof. Mag.rer.nat. Dr.rer.nat. Gerald Kuba
gerald.kuba@boku.ac.at
Tel: +43 1 47654-83514
Project Staff
01.02.2009 - 31.01.2012
BOKU partners
Funder
