Abstract:

We discuss different notions of uniformity or randomness of sets of integers or integer points, as well as their applications to some fundamental problems. Starting with the original proof of Roth on the existence on 3-term arithmetic progressions in subsets of the integers of positive density, we sketch the so-called higher degree uniformity norms of Gowers which were crucial in the recent development of additive number theory. Along the way we'll mention some open problems and analogous concepts in combinatorics and in the finite field settings.