# Critical Dimensions for counting Lattice Points in Euclidean Annuli

Mathematical Modelling of Natural Phenomena (2010)

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

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topParnovski, 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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- L. Parnovski, A. V. Sobolev. On the Bethe–Sommerfeld conjecture for the polyharmonic operator. Duke Math. J., 107 (2001), No. 2, 209–238. Zbl1092.35025
- L. Parnovski, A. V. Sobolev. Lattice points, perturbation theory and the periodic polyharmonic operator. Annales H. Poincaré, 2 (2001), 573–581. Zbl1162.35434
- M. Skriganov. Geometrical and arithmetical methods in the spectral theory of the multi-dimensional periodic operators. Proc. Steklov Math. Inst. Vol., (1984) 171.
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