Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T23:58:22.883Z Has data issue: false hasContentIssue false
  • English
  • Français

DISTRIBUTION OF INTEGER LATTICE POINTS IN A BALL CENTRED AT A DIOPHANTINE POINT

Part of: Additive number theory; partitions Harmonic analysis in several variables

Published online by Cambridge University Press:  10 December 2009

Hyunsuk Kang  and
Alexander V. Sobolev
Hyunsuk Kang
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, U.K. (email: [email protected])
Alexander V. Sobolev
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, U.K. (email: [email protected])
Get access

Abstract

We study the variance of the fluctuations in the number of lattice points in a ball and in a thin spherical shell of large radius centred at a Diophantine point.

MSC classification

Secondary: 11P21: Lattice points in specified regions 42B05: Fourier series and coefficients
Type
Research Article
Information
Mathematika , Volume 56 , Issue 1 , January 2010 , pp. 118 - 134
Copyright
Copyright © University College London 2010

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

[1]Bleher, P. and Bourgain, J., Distribution of the error term for the number of lattice points inside a shifted ball. In Analytic Number Theory, Vol. 1 (Allerton Park, IL, 1995) (Progress in Mathematics 3), Birkhäuser (Boston, 1996), 141–153.CrossRefGoogle Scholar
[2]Bleher, P. M., Cheng, Z., Dyson, F. J. and Lebowitz, J. L., Distribution of the error term for the number of lattice points inside a shifted circle. Comm. Math. Phys. 154(3) (1993), 433–469.CrossRefGoogle Scholar
[3]Bleher, P. and Dyson, F., Mean square limit for lattice points in a sphere. Acta Math. 67(3) (1994), 461–481.Google Scholar
[4]Bleher, P. and Dyson, F., Mean square value of exponential sums related to representation of integers as sum of two squares. Acta Math. 68(3) (1994), 71–84.Google Scholar
[5]Bleher, P. and Lebowitz, J. L., Variance of number of lattice points in random narrow elliptic strip. Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), 27–58.Google Scholar
[6]Heath-Brown, D. R., The distribution and moments of the error term in Dirichlet divisor problem. Acta Arith. 60 (1992), 389–415.Google Scholar
[7]Hughes, C. P. and Rudnick, Z., On the distribution of lattice points in thin annuli. Int. Math. Res. Not. 13 (2004), 637–658.CrossRefGoogle Scholar
[8]Jarnik, V., Über die Mittelwertsätze der Gitterpunktlehre. V. Časopis Pěst. Mat. Fys. 69 (1940), 148–174.Google Scholar
[9]Marklof, J., The Berry–Tabor conjecture. In Proceedings of the 3rd European Congress of Mathematics (Barcelona, 2000) (Progress in Mathematics 202), Birkhäuser (Basel, 2001), 421–427.CrossRefGoogle Scholar
[10]Marklof, J., Pair correlation densities of inhomogeneous quadratic forms, II. Duke Math. J. 115 (2002), 409–434.CrossRefGoogle Scholar
[11]Marklof, J., Pair correlation densities of inhomogeneous quadratic forms. Ann. of Math. (2) 158 (2003), 419–471.CrossRefGoogle Scholar
[12]Marklof, J., Mean square value of exponential sums related to the representation of integers as sums of squares. Acta Arith. 117 (2005), 353–370.CrossRefGoogle Scholar
[13]Schmidt, W. M., Approximation to algebraic numbers. Enseign. Math. (2) 17 (1971), 187–253.Google Scholar
[14]Wigman, I., The distribution of lattice points in elliptic annuli. Q. J. Math. 57 (2006), 395–423.CrossRefGoogle Scholar