We use the following notation: $$[N]:=\{1,2,3,\cdots,N\}$$ $$\mathbb{E}_{n\in [N]}f(x) : = \frac{1}{N}\sum_{n \in [N]}f(x)$$
This doubt is from the book titled Higher Order Fourier Analysis by T. Tao. The doubt is from the proof of proposition 1.1.13. The book states that:
We have that $$\left \vert \mathbb{E}_{n\in [N] } f(x_n)\right \vert >\delta.$$ Now, using the kernel bounds $$\int_{\mathbb{T}^d}K_R=1$$ where $K_R$ is the Fejer kernel $$K_R(x_1,\cdots,x_d):=\prod_{j=1}^{d}\frac{1}R\left(\frac{\sin(\pi R x_j)}{sin(\pi x_j)} \right)^2$$ and \begin{equation} \tag{1}|K_R(x)|\ll_d \prod_{j=1}^{d}R(1+R ||x_j||_{\mathbb{T}})^{-2},\end{equation} (where $f\ll_d g$ means that $f\leq C_d g$ where the constant $C$ depends on $d$ and $||\cdot||$ means the nearest distance to an integer ) and the Lipschitz nature of $f$, we see that \begin{equation}\tag{2}F_Rf(x) = f(x) + O_d(1/R).\end{equation}
Thus, if we choose $R$ to be a sufficiently small multiple of $\frac{1}{\delta}$, we get $$\Big\vert\mathbb{E}_{n \in [N]}F_Rf(x_n) \Big\vert \gg \delta$$ and thus by pigeonhole principle we have \begin{equation}\tag{3} \Big\vert\mathbb{E}_{n \in [N]}e(k\cdot x_n) \Big\vert \gg_d \delta^{O_d(1)}\end{equation} for some non-zero $k$ of magnitude $|k| \ll_d \delta^{-O_d(1)} .$
i) How does $(1)$ and $(2)$ follow?
ii) How does $(3)$ follow from pigeonhole principle and the fact that $\Big\vert\mathbb{E}_{n \in [N]}F_Rf(x_n) \Big\vert \gg \delta$?
My idea is the following:
$\Big\vert\mathbb{E}_{n \in [N]}F_Rf(x_n) \Big\vert \gg \delta \implies \Big\vert\mathbb{E}_{n \in [N]}\sum_{k \in \mathbb{Z}^d }m_R(k)\hat{f}(k)e(k \cdot x) \Big\vert \gg \delta $
Now, using the trivial bound $\hat{f}(k) = O(1)$ and $\hat{f}(0) =0$...
But how do we get $\delta$ raised to some power $(O_d(1))$ and how to use pigeonhole principle?