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This question is related to this and this ones. The Gauss Circle problem asks for the number $N(r)$ of integer points within a sphere of radius $r$ centered at the origin. It is well known that $N(r) …

This is called the Minkowski-Weyl Theorem (see, e.g., Zieglers Book). The proof of the "main theorem" essentially gives an algorithm to do that.

Let $\Lambda \subset \mathbb{R}^n$ be an Euclidean lattice with generator matrix $B$. Define the center density $\delta(\Lambda)$ in the usual way as $\delta(\Lambda) = \rho^n/|\det{B}|$, where $\rho$ …

The product $d\times d^*$ cannot be "so" large, as a consequence of the so-called Transference theorems Particularly, Thm. 2.1. of the paper shows that $d\times d^* \leq n^2$ (hence in your example, …

The two-dimensional version of this question was already asked (although in a different language) here. In fact, besides the generalization to measurable sets mentioned by Pete, this result can be ge …