Some new results on the higher energies

Recently, Kelley–Meka made a remarkable breakthrough by proving that any set A in {1,2,...,N} having no arithmetic progressions of length three has size at most N exp(-O((log N)^c)) for an absolute constant c>0. First, we discuss this result and, second, we obtain a generalization of this theorem that allows us to find applications to the so-called corners problem. In our proof we develop the theory of the higher energies. Also, we discuss the case of longer arithmetic progressions, as well as a general family of norms, which includes the higher energies norms and Gowers norms.
ZOOM meeting ID: 918 2692 4661
Passcode: a six digit number $N=(4!)^2+(p-5)^2$ where $p$ is the smallest prime such that $p>600$.