Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-19T06:42:19.788Z Has data issue: false hasContentIssue false
  • English
  • Français

Lp and Weak–Lp estimates for the number of integer points in translated domains

Published online by Cambridge University Press:  30 September 2015

LUCA BRANDOLINI ,
LEONARDO COLZANI ,
GIACOMO GIGANTE  and
GIANCARLO TRAVAGLINI
LUCA BRANDOLINI
Affiliation:
Dipartimento di Ingegneria Gestionale, dell'Informazione e della Produzione, Università di Bergamo, Viale Marconi 5, 24044 Dalmine BG, Italy. e-mail: luca.brandolini@unibg.it
LEONARDO COLZANI
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy. e-mail: leonardo.colzani@unimib.it
GIACOMO GIGANTE
Affiliation:
Dipartimento di Ingegneria Gestionale, dell'Informazione e della Produzione, Università di Bergamo, Viale Marconi 5, 24044 Dalmine BG, Italy. e-mail: giacomo.gigante@unibg.it
GIANCARLO TRAVAGLINI*
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy.
*
e-mail: giancarlo.travaglini@unimib.it
Get access Rights & Permissions [Opens in a new window]

Abstract

Revisiting and extending a recent result of M. Huxley, we estimate the Lp($\mathbb{T}$d) and Weak–Lp($\mathbb{T}$d) norms of the discrepancy between the volume and the number of integer points in translated domains.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 3 , November 2015 , pp. 471 - 480
Copyright
Copyright © Cambridge Philosophical Society 2015 

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

REFERENCES

[1] Bergh, J. and Löfström, J. Interpolation Spaces (Spinger Verlag, 1976).Google Scholar
[2] Bonnesen, T. and Fenchel, W. Theory of Convex Bodies (BCS Associates, Moscow, Idaho, USA, 1987).Google Scholar
[3] Brandolini, L., Gigante, G. and Travaglini, G. Irregularities of distribution and average decay of Fourier transforms. In Chen, W., Srivastav, A., Travaglini, G. (eds.), A Panorama of Discrepancy Theory, Lecture Notes in Math. 2107 (Springer International Publishing, Switzerland, 2014).Google Scholar
[4] Brandolini, L., Hofmann, S. and Iosevich, A. Sharp rate of average decay of the Fourier transform of a bounded set. Geom. Funct. Anal. 13 (2003), 671–680.Google Scholar
[5] Brandolini, L., Rigoli, M. and Travaglini, G. Average decay of Fourier transforms and geometry of convex sets. Rev. Mat. Iberoamericana 14 (1998), 519–560.Google Scholar
[6] Bruna, J., Nagel, A. and Wainger, S. Convex hypersurfaces and Fourier transforms. Ann. of Math. 127 (1988), 333–365.Google Scholar
[7] Gelfand, I. M., Graev, M. I. and Vilenkin, N. Y. Generalised Functions, Vol. 5: Integral Geometry and Problems of Representation Theory (Academic Press, New York, 1966).Google Scholar
[8] Herz, C. On the number of lattice points in a convex set. Amer. J. Math. 84 (1962), 126–133.Google Scholar
[9] Hlawka, E. Uber Integrale auf convexen Körpen, I, II. Monatsh. Math. 54 (1950), 1–36, 81–99.Google Scholar
[10] Huxley, M. N. A fourth power discrepancy mean. Monatsh. Math. 73 (2014), 231–238.Google Scholar
[11] Kendall, D. G. On the number of lattice points inside a random oval. Quart. J. Math. Oxford 19 (1948), 1–26.Google Scholar
[12] Kolountzakis, M. N. and Wolff, T. On the Steinhaus tiling problem. Mathematika 46 (1999), 253–280.Google Scholar
[13] Parnovski, L. and Sobolev, A. V. On the Bethe–Sommerfeld conjecture for the polyharmonic operator. Duke Math. J. 107 (2001), 209–238.Google Scholar
[14] Podkorytov, A. N. On the asymptotics of the Fourier transform on a convex curve. Vestnik Leningrad University Mathematics 24 (1991), 57–65.Google Scholar
[15] Stein, E. M. Singular Integrals and Differentiabilty Properties of Functions (Princeton University Press, 1970).Google Scholar
[16] Stein, E. M. Harmonic Analysis (Princeton University Press, 1993).Google Scholar
[17] Stein, E. M. and Weiss, G. Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, 1971).Google Scholar
[18] Travaglini, G. Number Theory, Fourier Analysis and Geometric Discrepancy (Cambridge University Press, 2014).Google Scholar