Olof Sisask: Using continuity to find solutions to linear equations in sets of integers

Abstract. An old question of Erdos and Turan asks how large a subset of {1,2,...,N} can be if it is not allowed to contain three elements in arithmetic progression, and for a while after it was asked it was completely unclear how one should approach this seemingly combinatorial or number theoretical question. Roth ingeniously used Fourier analysis to show that the answer is at most o(N), indeed O(N/log log N), but to this day massive gaps remain between the best upper and lower bounds known for this quantity, which in the end seems to be intimately tied to questions in analysis. Three-term arithmetic progressions correspond to solutions to the equation x+y=2z, and until recently a similar situation existed for almost all equations in four or more variables whose coefficient-sum is 0. Here we shall use a mix of Fourier-analytic and non-Fourier methods to show that the upper and lower bounds are not very far apart for the analogous question for the equation x+y+z=3w. The key result is analytic: it says that three-fold convolutions of certain kinds of functions are (quantitatively) uniformly continuous in a particular topology -- the Bohr topology.
I plan to describe this result and say how it fits into the above picture, assuming fairly little previous knowledge of the area. Based on joint work with Tomasz Schoen.