Geodesic
Asymptotics of variance of the lattice point count
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 3, pp. 751-758 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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The variance of the number of lattice points inside the dilated bounded set $rD$ with random position in $\Bbb R^d$ has asymptotics $\sim r^{d-1}$ if the rotational average of the squared modulus of the Fourier transform of the set is $O(\rho ^{-d-1})$. The asymptotics follow from Wiener's Tauberian theorem.
The variance of the number of lattice points inside the dilated bounded set $rD$ with random position in $\Bbb R^d$ has asymptotics $\sim r^{d-1}$ if the rotational average of the squared modulus of the Fourier transform of the set is $O(\rho ^{-d-1})$. The asymptotics follow from Wiener's Tauberian theorem.
Classification : 11H06, 62D05
Keywords: point lattice; Fourier transform; volume; variance 
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Janáček, Jiří. Asymptotics of variance of the lattice point count. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 3, pp. 751-758. http://geodesic.mathdoc.fr/item/CMJ_2008_58_3_a12/

[1] Bochner, S., Chandrasekharan, K.: Fourier transforms. Princeton University Press (1949). | MR | Zbl

[2] Brandolini, L., Hofmann, S., Iosevich, A.: Sharp rate of average decay of Fourier transform of a bounded set. Geom. Func. Anal. 13 (2003), 671-680. | DOI | MR

[3] Janáček, J.: Variance of periodic measure of bounded set with random position. Comment. Math. Univ. Carolinae 47 (2006), 473-482. | MR

[4] Kendall, D. G.: On the number of lattice points inside a random oval. Quarterly J. Math. 19 (1948), 1-26. | DOI | MR | Zbl

[5] Kendall, D. G., Rankin, R. A.: On the number of points of a given lattice in a random hypersphere. Quarterly J. Math., 2nd Ser. 4 (1953), 178-189. | DOI | MR | Zbl

[6] Matérn, B.: Precision of area estimation: a numerical study. J. Microsc. 153 (1989), 269-283. | DOI

[7] Matheron, G.: Les variables regionalisés et leur estimation. Masson et CIE, Paris (1965).

[8] Rao, R. C.: Linear statistical inference and its applications. 2nd ed. , John Wiley & Sons, New York (1973). | MR | Zbl

[9] Rataj, J.: On set covariance and three-point test sets. Czech. Math. J. 54 (2004), 205-214. | DOI | MR | Zbl

[10] Watson, G. N.: A treatise on the theory of Bessel functions. 2nd edition, Cambridge University Press (1922). | MR

[11] Rudin, W.: Functional Analysis. McGraw-Hill Book Company (1973). | MR | Zbl

[12] Varchenko, A.: Number of lattice points in families of homothetic domains in $\Bbb R^n$. Func. Anal. Appl. 17 (1983), 79-83. | DOI | MR

[13] Wiener, N.: The Fourier integral and certain of its applications. Dover Publications Inc., New York (1933). | MR | Zbl