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I am informed that you are "counting lattice points inside of a polyhedron."

Here is a lecture on the subject - the picture on page six looks like the version of the problem you are interested in. To be honest, I found these notes by doing a google search. I am told that this is a huge field!

It might help if you could narrow your problem even further. For example, you say that the $x_i$ are bounded real numbers. Do you know these to some high precision? Or can you give some information on how the $x_i$ are given? And can you say the same for $S$?

EDIT: Here is a survey paper by the same author, Jesús De Loera, covering the same material in greater detail.

It might help if you could narrow your problem even further. For example, you say that the $x_i$ are bounded real numbers. Do you know these to some high precision? Or can you give some information on how the $x_i$ are given? And can you say the same for $S$?