Geodesic
On the number of integer points in a multidimensional domain
Diskretnaya Matematika, Tome 29 (2017) no. 4, pp. 106-120

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We provide a new upper estimate for the modulus of the difference $|\Lambda\cap {\cal S}|-{\rm vol }_n({\cal S})/{\rm det }\,\Lambda$, where ${\cal S}\subset \mathbb R^n$ is a set of volume ${\rm vol }_n({\cal S})$ and $\Lambda\subset \mathbb R^n$ is a complete lattice with determinant ${\rm det }\,\Lambda$. This result has an important practical application, for example, in estimating the number of integer solutions of an arbitrary system of linear and nonlinear inequalities.
Keywords: integer lattice, number of integer points, Gaussian volume heuristic. 
@article{DM_2017_29_4_a6,
     author = {A. S. Rybakov},
     title = {On the number of integer points in a multidimensional domain},
     journal = {Diskretnaya Matematika},
     pages = {106--120},
     publisher = {mathdoc},
     volume = {29},
     number = {4},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2017_29_4_a6/}
}
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A. S. Rybakov. On the number of integer points in a multidimensional domain. Diskretnaya Matematika, Tome 29 (2017) no. 4, pp. 106-120. http://geodesic.mathdoc.fr/item/DM_2017_29_4_a6/