Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Planar convex bodies, Fourier transform, lattice points, and irregularities of distribution


Authors: L. Brandolini, A. Iosevich and G. Travaglini
Journal: Trans. Amer. Math. Soc. 355 (2003), 3513-3535
MSC (2000): Primary 42B10; Secondary 52A10
Published electronically: April 25, 2003
MathSciNet review: 1990161
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $B$ be a convex body in the plane. The purpose of this paper is a systematic study of the geometric properties of the boundary of $B$, and the consequences of these properties for the distribution of lattice points in rotated and translated copies of $\rho B$ ($\rho$ being a large positive number), irregularities of distribution, and the spherical average decay of the Fourier transform of the characteristic function of $B$. The analysis makes use of two notions of ``dimension'' of a convex set. The first notion is defined in terms of the number of sides required to approximate a convex set by a polygon up to a certain degree of accuracy. The second is the fractal dimension of the image of the Gauss map of $B$. The results stated in terms of these quantities are essentially sharp and lead to a nearly complete description of the problems in question.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42B10, 52A10

Retrieve articles in all journals with MSC (2000): 42B10, 52A10


Additional Information

L. Brandolini
Affiliation: Dipartimento di Ingegneria, Università di Bergamo, Viale G. Marconi 5, 24044 Dalmine (BG), Italy
Email: brandolini@unibg.it

A. Iosevich
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri
Email: iosevich@math.missouri.edu

G. Travaglini
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy
Email: travaglini@matapp.unimib.it

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03240-9
PII: S 0002-9947(03)03240-9
Keywords: Decay of Fourier transforms, convex bodies, Minkowski dimension, lattice points, irregularities of distribution
Received by editor(s): February 11, 2002
Published electronically: April 25, 2003
Additional Notes: The first and third authors are supported by MURST. The second author is supported by NSF grant DMS00-87339
Article copyright: © Copyright 2003 American Mathematical Society