Real Algebraic Geometry in Additive Number Theory

During the last decade new techniques have been developed in order to solve polynomial optimization problems that are invariant under the action of a group using semidefinite programming. In this thesis we apply these methods to additive number theory. In particular we set up a novel machinery for counting arithmetic progressions using real algebraic geometry. We prove several new results related to Szemerédi's theorem. We prove results for arithmetic progressions of length 3 that one has not been able to prove using Fourier analytic methods or ergotic theory, which are the most commonly used tools in additive number theory. Instead of trying to prove existence of arithmetic progressions we do the most natural generalization: we count them. To our knowledge we give the first results on the form "There are at least W(k,G,d) arithmetic progressions of length k in any subset S of the elements of G with |S|=|G|d.", and we discuss how our results could possibly be improved in order to give a new proof of Szemerédi's theorem using real algebraic geometry. A similar type of results are theorems on the form "There are at least R(k,G,c) monochromatic arithmetic progressions of length k in any c-coloring of the finite group G.". There are many theorems of this type for 2-colorings, and we prove several new results in this thesis. Some of the new results hold for any finite group G, including non-abelian groups, which are very difficult to analyze using Fourier analytic methods.