ABSTRACT: We start from two basic theorems in harmonic analysis on Euclidean spaces emphasizing the role of curvature, and formulate their discrete analogue, where the underlying space is the set of integer points or its dual the flat torus.

We intend to explain how these in turn imply results of various kinds; such as $L^p$-bounds on the eigenfunctions of the Laplacian on the torus, and ergodic theorems showing the equidistribution of solutions of diophantine equations when mapped into measure spaces via an ergodic family of transformations.

In the proofs properties of certain exponential sums are crucial and exploited by methods of analytic number theory, such as the Hardy-Littlewood method.