I've got something like a "meta-question": In some parts of combinatorics, looking at Fourier transforms can be a very helpful tool. For example, an early proof of Roth's theorem (any sufficiently large subset $A$ of $\{1,2,\dots,n\}$ of some given density contains an arithmetic progression of length $3$ (or, more general, length $k$, but then the proof I'm going to briefly sketch won't work)) builds on using Fourier transforms. Essentially, you prove that either $A$ contains an AP of length $3$ or there exists some $r$ such that $\hat A(r)$ is fairly large (where $A$ is also used to denote the indicator function of the set $A$ and $\hat A$ denotes its Fourier transform, i.e. $\hat A(r)=\mathbb{E}f(x)\omega^{-rx}$). You then go on to show that the latter implies that there exists an AP $A'$ of $\{1,2,\dots,n\}$, which is fairly long and moreover, when viewed as a subset of $A$, it larger density and this eventually has to stop.
Now, my question is: How should I think about $A$ having a large Fourier coefficient as in the proof? What does this tell me?
A better way to phrase the case distinction is as follows: Either the set $A$ is random-like, or not.
If $A$ has small Fourier transforms, turns out we can essentially pretend that $A$ is a random subset of $[n]$ of size $|A|$. In this case, $A$ contains many $3$-APs, as one would expect from a random set. Otherwise, for some $r$, $\hat{A}(r)$ is large. Turns out that this implies that $A$ meets a long arithmetic progression in way more elements than we would expect had $A$ been a random-like set. Hence, the statement that $\hat{A}(r)$ is large can be interpreted as saying $A$ is not-random-like in this sense.