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

Published online by Cambridge University Press:  10 December 2009

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])
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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.

Type
Research Article
Copyright
Copyright © University College London 2010

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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), 141153.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), 433469.CrossRefGoogle Scholar
[3]Bleher, P. and Dyson, F., Mean square limit for lattice points in a sphere. Acta Math. 67(3) (1994), 461481.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), 7184.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), 2758.Google Scholar
[6]Heath-Brown, D. R., The distribution and moments of the error term in Dirichlet divisor problem. Acta Arith. 60 (1992), 389415.Google Scholar
[7]Hughes, C. P. and Rudnick, Z., On the distribution of lattice points in thin annuli. Int. Math. Res. Not. 13 (2004), 637658.CrossRefGoogle Scholar
[8]Jarnik, V., Über die Mittelwertsätze der Gitterpunktlehre. V. Časopis Pěst. Mat. Fys. 69 (1940), 148174.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), 421427.CrossRefGoogle Scholar
[10]Marklof, J., Pair correlation densities of inhomogeneous quadratic forms, II. Duke Math. J. 115 (2002), 409434.CrossRefGoogle Scholar
[11]Marklof, J., Pair correlation densities of inhomogeneous quadratic forms. Ann. of Math. (2) 158 (2003), 419471.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), 353370.CrossRefGoogle Scholar
[13]Schmidt, W. M., Approximation to algebraic numbers. Enseign. Math. (2) 17 (1971), 187253.Google Scholar
[14]Wigman, I., The distribution of lattice points in elliptic annuli. Q. J. Math. 57 (2006), 395423.CrossRefGoogle Scholar