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Critical Dimensions for counting Lattice Points in Euclidean Annuli

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 the annulus. The average number of lattice points there equals the volume of the annulus, and we study the L1 and L2 norms of the remainder. We say that a dimension is critical, if these norms do not have upper and lower 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 of the width of an annulus as a function of the radius determines which dimensions are critical now. In particular, if the width is bounded away from zero and infinity, the critical dimensions are d ≡ 3 mod 4; if the width goes to infinity, but slower than the radius, then all dimensions are critical, and if the width tends to zero as 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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