19. Roth's theorem II Fourier analytic proof in the integers
[SQUEAKING] [PAPER RUSTLING] [CLICKING] YUFEI ZHAO: Last time we started talking about Roth's theorem, and we showed a Fourier analytic proof of Roth's theorem in the finite field model. So Roth's theorem in F3 to the N. And I want to today show you how to modify that proof to work in integers. And this will be basically Roth's original proof of his theorem. OK. So what we'll prove today is the statement that the size of the largest 3AP-free subset of 1 through N is, at most, N divided by log log N. OK, so we'll prove a bound of this form. The strategy of this proof will be very similar to the one that we had from last time. So let me review for you what is the strategy. So from last time, the proof had three main steps. In the first step, we observed that if you are in the 3AP-free set then there exists a large Fourier coefficient. From this Fourier coefficient, we were able to extract a large subspace where there is a density increment. I want to...