Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-14T23:27:45.319Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometry of the gauss map and lattice points in convex domains

Part of: Discrete geometry

Published online by Cambridge University Press:  26 February 2010

L. Brandolini ,
L. Colzani ,
A. Iosevich ,
A. Podkorytov  and
G. Travaglini
L. Brandolini
Affiliation:
Dipartimento di Ingegneria, Gestionale e de l'lnformazione, Universita degli Studi di Bergamo, Viale Marconi 5, 24044 Dalmine (BG), Italy E-mail: brandolini@unibg.it
L. Colzani
Affiliation:
Dipartimento di Matematica e Applicationi, Universita di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy E-mail: leonardo@matapp.unimib. it
A. Iosevich
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.http://www.math.missouri.edu/-iosevich/ E-mail: iosevich@math.missouri.edu.
A. Podkorytov
Affiliation:
Department of Mathematics, St. Petersburg State University, Bibliotecnaya pi. 2, Peterhof, St. Petersburg, 198904, Russia E-mail: pan@pan.usr.pu.ru.
G. Travaglini
Affiliation:
Dipartimento di Matematica e Applicazioni, Universita di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italyhttp://www.matapp.unimib.it/~travaglini/ E-mail: travaglini@matapp.unimib.it.
Get access

Abstract

Let Ω be a convex planar domain, with no curvature or regularity assumption on the boundary. Let Nθ(R) = card{RΩθ∩ℤ2}, where Ωθ denotes the rotation of Ω by θ. It is proved that, up to a small logarithmic transgression, Nθ(R) = |Ω|R2 + O(R2/3), for almost every rotation. A refined result based on the fractal structure of the image of the boundary of Ω under the Gauss map is also obtained.

MSC classification

Secondary: 52C05: Lattices and convex bodies in $2$ dimensions
Type
Research Article
Information
Mathematika , Volume 48 , Issue 1-2 , December 2001 , pp. 107 - 117
Copyright
Copyright © University College London 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[BIT]Brandolini, L., Iosevich, A. and Travaglini, G., Planar convex bodies, Fourier transform, lattice points, and irregularities of distributions, Trans. Amer. Math. Soc. (to appear).Google Scholar
[BRT]Brandolini, L., Rigoli, M. and Travaglini, G., Average decay of Fourier transforms and geometry of convex sets, Revista Mat. Iber. 14 (1998), 519–560.CrossRefGoogle Scholar
[C]Colin de Verdiere, Y., Nombre de points entiers dans une famille homothetique de domains de R, Ann. Sci. Ecole Norm. Sup. 10 (1977), 559–575.CrossRefGoogle Scholar
[He]Herz, C. S., On the number of lattice points in a convex set, Amer. J. Math. 84 (1962), 126–133.CrossRefGoogle Scholar
[HI]Hlawka, E., Uber Integrale auf konvexen Korpen, I & II, Monatsh, Math. 54 (1950), 1– 36, 81-99.CrossRefGoogle Scholar
[Hu]Huxley, M. N., Area, Lattice Points, and Exponential Sums. Clarendon Press, Oxford (1996).CrossRefGoogle Scholar
[I]Iosevich, A., Lattice points and generalized diophantine conditions, J. Number Theory 90 (2001), 19–30.CrossRefGoogle Scholar
[J]Jarnik, V., Uber die Gitterpunkte auf Konvexen Kurven, Math. Zeitschrift 24 (1925), 500–518.CrossRefGoogle Scholar
[L]Landau, E., Uber die Gitterpunkte in einem Kreise (II), Gottinger Nachrichten (1915), 161–171.Google Scholar
[P]Podkorytov, A. N., The asymptotic of a Fourier transform on a convex curve, Vestn. Leningr. Univ. Mat. 24 (1991), 57–65.Google Scholar
[R]Randol, B., A lattice-point problem II, Trans. Amer. Math. Soc. 125 (1966), 101–113.CrossRefGoogle Scholar
[RN]Riesz, F. and Sz-Nagy, B., Functional Analysis, Frederick Ungar Publishing, New York (1978).Google Scholar
[Sv]Svensson, I., Estimates for the Fourier transform of the characteristic function of a convex set, Ark. Mat. 9 (1971), 11–22.CrossRefGoogle Scholar
[Sk]Skriganov, M. M., Ergodic theory on SL(n), Diophantine approximations and anomalies in the lattice point problem, Invent. Math. 132 (1998), 1–72.CrossRefGoogle Scholar