So I've been reading through [On the number of monochromatic Schur triples](https://www.sciencedirect.com/science/article/pii/S0196885803000101) by Datrovsky on finding the minimal number of Schur triples. This means you're trying to 2-colour the set of the smallest n positive integers in such a way as to minimize the number of monochromatic triples (a,b,c) where a+b=c. 


In the first paragraph or so he somehow gets a formula for these with something discrete Fourier-esque (called equation (1.1) in the paper). With a lot of algebra I think I've managed to convince myself that this formula holds but I'm not sure how one would think of this or when it would generalize. In particular I don't follow the steps in corollary 1 where he generalizes to Schur k-tuples. It also seems like the same trick is reused in a few places in the paper so I'd really like to get a handle on exactly what tools are being used here. 


If anyone could give some insight I'd really appreciate it. 



Schur triples question
Fourier analysis is a well-established tool in additive combinatorics,
and counting questions are routinely translated into equivalent questions about the Fourier transform.
This survey of Gowers might give you an idea of the breadth and scope: <https://arxiv.org/abs/1608.04127>.


