Geodesic
Asymptotics of variance of the lattice point count
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 3, pp. 751-758

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MR Zbl
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 
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/
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