On the number of integer points in a multidimensional domain

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.