This question arises from the book titled 'Higher order fourier analysis' by T. Tao.
Fourier complexity: A function $f:[N] \to \mathbb{C}$ has fourier complexity at most M if it can be expressed as $$f(n) =\sum_{m=1}^M'c_me(\alpha_m n)$$ for some $M' \leq M$ and some complex numbers $c_1,\cdots,c_{M'}$ of magnitude atmost $1$, and some real numbers $\alpha_1, \cdots, \alpha_{M'}$.
Also, if $f,g$ has fourier complexity $M$, then $f+g, f-g, \overline{f},or fg$ all have Fourier complexity atmost $O_M(1)$.
Definition: Let $\mathcal{F}: \mathbb{R}^{+} \to \mathbb{R}^{+}$ be a function. We say that a function $f:[N]\to \mathbb{C}$ is Fourier Measurable with growth function $\mathcal{F}$ if, for every $K>1$, one can find a function $f_k:[N]\to \mathbb{C}$ of Fourier complexity at most $\mathcal{F}(K)$ such that $\mathbb{E}_{n\in [N]}|f(n)-f_K(n)|\leq 1/K$.
A subset $A$ of $[N]$ is fourier measurable with growth function $\mathcal{F}$ if $1_A$ is fourier measurable with this growth function.
Question: I wish to know that how does a fourier measurable set look like? What role or significance does it have?
Question: Can someone please give me an example of a fourier measurable set and a fourier measurable function (except the indicator function of the set). I want to grasp the intuitive idea behind introducing such a definition and why is it so useful!
Prof Terence Tao uses this definition in a lemma which is stated below:
Lemma: (correlation with fourier character implies correlation with a fourier measurable set). Let $f:[N]\to \mathbb{C}$ be bounded in magnitude by $1$, and suppose that $|\mathbb{E}_{n\in [N]} f(n) e(-\alpha n)|\geq \delta$ for some $\delta >0$. Then there exists a fourier-measurable set $E\subset [N]$ with growth function $\mathcal{F}$ depending on $\delta$ such that $|\mathbb{E}_{n\in [N]}f(n)1_E(n)|\gg \delta$.