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Critical Dimensions for counting Lattice Points in EuclideanAnnuli

Published online by Cambridge University Press:  12 May 2010

L. Parnovski  and
N. Sidorova
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Abstract

We study the number of lattice points in ℝd ,d ≥ 2, lying inside an annulus as a function of the centre of theannulus. The average number of lattice points there equals the volume of the annulus, andwe study the L 1 and L 2 norms ofthe remainder. We say that a dimension is critical, if these norms do not have upper andlower bounds of the same order as the radius goes to infinity. In [Duke Math. J., 107 (2001), No. 2, 209–238], it was proved that in the case of the ball (instead of an annulus)the critical dimensions are d ≡ 1 mod 4. We show that the behaviour ofthe width of an annulus as a function of the radius determines which dimensions arecritical now. In particular, if the width is bounded away from zero and infinity, thecritical dimensions are d ≡ 3 mod 4; if the width goes to infinity, butslower than the radius, then all dimensions are critical, and if the width tends to zeroas a power of the radius, then there are no critical dimensions.

Keywords

lattice points
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 4: Spectral problems. Issue dedicated to the memory of M. Birman , 2010 , pp. 293 - 316
Copyright
© EDP Sciences, 2010

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References

G. E. Andrews, R. Askey, R. Roy. Special Functions. Cambridge University Press, 1999.
Cheng, Z., Lebowitz, J. L., Major, P.. On the number of lattice points between two enlarged and randomly shifted copies of an oval . Probab. Theory Rel. Fields, 100 (1994), No. 2, 253–268.CrossRefGoogle Scholar
Götze, F.. Lattice point problems and values of quadratic forms . Invent. Math., 157 (2004), No. 1, 195–226.CrossRefGoogle Scholar
Huxley, M.N.. Exponential sums and lattice points. III . Proc. London Math. Soc., (3) 87 (2003), No. 3, 591–609. CrossRefGoogle Scholar
Kendall, D. G.. On the number of lattice points inside a random oval . Quart. J. Math., Oxford Ser. 19, (1948), 1–26. CrossRefGoogle Scholar
Parnovski, L., Sobolev, A. V.. On the Bethe–Sommerfeld conjecture for the polyharmonic operator . Duke Math. J., 107 (2001), No. 2, 209–238.Google Scholar
Parnovski, L., Sobolev, A. V.. Lattice points, perturbation theory and the periodic polyharmonic operator . Annales H. Poincaré, 2 (2001), 573–581.CrossRefGoogle Scholar
M. Skriganov. Geometrical and arithmetical methods in the spectral theory of the multi-dimensional periodic operators. Proc. Steklov Math. Inst. Vol., (1984) 171.
A. Walfisz. Gitterpunkte in mehrdimensionalen Kugeln. Warszawa: Panstwowe Wydawnictwo Naukowe, 1957.