Suppose one did have an exact formula for the number of $\mathbb{Z}^n$-lattice points intersecting an arbitrary dilate of a (not necessarily rational) finite, closed and convex $n$-polytope. As a result, a formula for the number of non-negative integral solutions of the inequality $\sum_{i = 1}^{n} q_i x_i \leq R$ for positive real coefficients and $R > 0$ would follow immediately, as well as, the de Bruijn function for the number of smooth numbers less than or equal to any given positive real greater than $1$. The solution of a generalization of Ramanujan's Triangle Problem of computing the exact number of positive integers of the form $1 \leq 2^a 3^b \leq N$ would then also follow.
What other interesting problems could be solved or, at least in principle, be better understood by solving the above lattice point enumeration problem? References are certainly welcome!
