Problems in additive and extremal combinatorics

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This research will focus on a cluster of related problems and concepts that belong to additive combinatorics, an area of mathematics at the interface between analytic number theory, extremal combinatorics, harmonic analysis, dynamical systems, and group theory. The common feature of the problems is that they tend to be of a combinatorial flavour, and fairly easy to state, and that the right approach to these problems is usually to bring in tools from other areas. Over the last twenty years or so, this has led to a very fruitful cross-fertilization, and also to a number of very interesting questions, some of which will be attempted during this thesis.