Critical Dimensions for counting Lattice Points in Euclidean Annuli

L. Parnovski; N. Sidorova

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 4, page 293-316
  • ISSN: 0973-5348

Abstract

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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.

How to cite

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Parnovski, L., and Sidorova, N.. "Critical Dimensions for counting Lattice Points in Euclidean Annuli." Mathematical Modelling of Natural Phenomena 5.4 (2010): 293-316. <http://eudml.org/doc/197703>.

@article{Parnovski2010,
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.},
author = {Parnovski, L., Sidorova, N.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {lattice points; critical dimensions; lattice points in Euclidean annuli},
language = {eng},
month = {5},
number = {4},
pages = {293-316},
publisher = {EDP Sciences},
title = {Critical Dimensions for counting Lattice Points in Euclidean Annuli},
url = {http://eudml.org/doc/197703},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Parnovski, L.
AU - Sidorova, N.
TI - Critical Dimensions for counting Lattice Points in Euclidean Annuli
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/5//
PB - EDP Sciences
VL - 5
IS - 4
SP - 293
EP - 316
AB - 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.
LA - eng
KW - lattice points; critical dimensions; lattice points in Euclidean annuli
UR - http://eudml.org/doc/197703
ER -

References

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

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