Counting Lattice Points in Real Polytopes

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!

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Can you make this problem community wiki? –  Gerry Myerson Mar 7 '11 at 5:41
Solving a very similar problem (sum of a function over a lattice polytope) would allow one to get decent approximations to uniform samples of short DNA sequences w/r/t their binding energy. See mathoverflow.net/questions/10493/… –  Steve Huntsman Mar 7 '11 at 11:06
This is community wiki, I vote to close otherwise. –  Igor Rivin Mar 7 '11 at 15:54
I replaced the "big-list" tag by the "convex-polytopes" tag. It is unlikely that the problem will attract enough answers to qualify as a big-list problem. –  Gil Kalai Apr 19 '11 at 9:14
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