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I'm preparing a talk on lattice point enumeration in polytopes (Ehrhart-Macdonald Theory), and I'd like to have an introduction with a few motivational problems/results which arise from the application of lattice point methods. (The audience consists of junior and senior faculty across the mathematical disciplines.) A few that immediately come to mind are the following: Pick's theorem, Ehrhart reciprocity, Minkowski's bound in algebraic number theory and Hilbert series in Stanley-Reisner rings.

Question: What kinds of interesting problems have been tackled (or generated some widespread interest) by counting points in lattices?

Thanks!

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You may well know this paper already but I think it is very nice:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.87.2267&rep=rep1&type=pdf

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