Given a somewhat large subset A of F_pⁿ, an n-dimensional vector space over A of F, must the sumset A+A+A contain a large subspace? Can we say anything interesting about A itself, based solely on its size? In this two-part talk I shall introduce Fourier analysis on finite abelian groups and show how this can be used to deal with questions such as these. This will involve a version of the notion of almost-periodicity, which we shall be able to translate into tangible combinatorial results. I plan to assume very few prerequisites.