Notation

I will use a lot of notation without really explaining it, and some of it is not standard. I will list them here as I think of or use them. If there is something unexplained, please comment and I’ll add it on here.

If $A$ is a set then I also use $A(x)$ for the characteristic function of $A$, so that $A(x)=1$ when $x\in A$ and $0$ otherwise. Hopefully it will always be clear from context whether I am talking about the set or the function.

If $G$ is a finite abelian group then we have the dual group of characters $\widehat{G}$, the set of homomorphisms from $G$ to the circle group $S^1$. Given a function $f:G\to\mathbb{C}$ we can define the Fourier transform $\widehat{f}:\widehat{G}\to\mathbb{C}$ as

$\widehat{f}(\gamma)=\mathbb{E}_{x\in G}f(x)\gamma(x).$