Geodesic
Lattice point problem and the question of estimation and detection of smooth functions of many variables
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 21, Tome 431 (2014), pp. 198-208 
I. A. Suslina. Lattice point problem and the question of estimation and detection of smooth functions of many variables. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 21, Tome 431 (2014), pp. 198-208. http://geodesic.mathdoc.fr/item/ZNSL_2014_431_a12/
@article{ZNSL_2014_431_a12,
     author = {I. A. Suslina},
     title = {Lattice point problem and the question of estimation and detection of smooth functions of many variables},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {198--208},
     year = {2014},
     volume = {431},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_431_a12/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We consider the problem of asymptotics of $N_d(m)$, where $N_d(m)$ is the number of integer lattice points in the $d$-dimensional ball of radius $m$ (in $l_1$ and $l_2$-norms) for $d\to\infty$, $m\to\infty$. We show that this asymptotics differs from the asymptotic volume of $d$-dimensional ball of radius $m$ when the rate of convergence of $d$ to infinity is sufficiently high in comparison with that of $m$.

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