In W. T. Gower's essay The Two Cultures of Mathematics, he mentions the following as an example of a 'general principle' in combinatorics:
"If one is counting solutions, inside a given set, to a linear equation, then it is enough, and usually easier, to estimate Fourier coefficients of the characteristic function of the set."
Can someone clarify this, or better yet give an example of this principle in action?
Very interesting paper. Your quote is on page 8; it seems likely to me that the theorem of Roth on the next page is a special case of what he means.
If so then we're talking about a set $S\subset\{1,.\dots,N\}$, a "solution to a linear equation" inside $S$ is something like $n,n+a,n+2a\in S$ (which is the same as saying there exist $x,y\in S$ with $(x+y)/2\in S$; there's the linear equation), and the "Fourier coefficients" in question are the coefficients on the group $\Bbb Z_N$.
Not that I see how the argument goes, but what he writes is clearly not intended to be anything more than a sketch of the proof.