Analysis Seminar: "Szemeredi's theorem on arithmetic progressions II: Fourier analysis approach"

Abstract: Any set of natural numbers with nonzero upper density contains arbitrary large (but finite) arithmetic progressions (Szemeredi, 1975). Equivalently, given natural number k and positive real number a there exists a natural number N such that any subset of the first N natural numbers with cardinality no less than aN contains an arithmetic progression with k terms. We explain Roth's Fourier analysis approach to prove this when k=3. The ingredients are a sieve method based on discrete Fourier transform combined with some Diophantine approximations.

Host: John McCarthy