At the end of the last post, we outlined an approach to proving Roth’s theorem via a density-increment method. I’ll re-post it here, a little more fleshed out, for convenience.
Sketch. The idea is to deal with “structured sets” (like sets that are themselves long arithmetic progressions) and “pseudo-random sets” in different ways; we saw last time that the density-increment argument looks promising for structured sets, but “random” sets won’t succumb to it.
First, we need to make rigorous our intuition that “random enough” sets of positive density will have length-3 APs. The obvious way to do this is by showing the contrapositive. We want to find some suitable “randomness measure” such that random dense sets have high randomness measure. Now let be a positive-density set without 3-APs; we want to show that this condition is enough to guarantee that has low randomness measure.
Now for our modified-probabilistic argument. We want to show that “low-randomness” sets can’t be uniformly distributed (mod d) for every d; this guarantees the existence of an AP on which it has higher density. We pass to this AP; rinse, lather, and repeat, until our density rises above some pre-set bound ( will do nicely.) We also want to introduce a tool that tells us how much a set “looks like” an arithmetic progression with common difference d; this is because we need to obtain quantitative bounds for the density-increment argument, in order to ensure that the densities of our set on the sequence of APs don’t monotonically approach, say, 0.0000001.
So we’re looking for two technical tools; one that quantifies the “randomness” of a set, and one that lets us correlate our set with a candidate arithmetic progression to pass to so that our density-increment argument will actually work quantitatively. As it turns out, though, we can find one tool that will do both jobs for us; the discrete Fourier transform.