Tao Lemma 3.1 (3.3): |sum F(n) e(alpha n)| <= ||F^(k)||_1 / |2 sin(pi alpha)|^k, all k >= 1

Let $F:\mathbb R\to\mathbb C$ be smooth and compactly supported, let $\alpha\in\mathbb R$ with $\sin(\pi\alpha)\neq 0$ (that is, $\alpha\notin\mathbb Z$), and write $e(t)=e^{2\pi i t}$. Then for every integer $k\ge 1$,

Here $F^{(k)}$ denotes the $k$-th derivative of $F$ and $\|g\|_{L^{1}(\mathbb R)}=\int_{\mathbb R}|g|$. The hypothesis $\sin(\pi\alpha)\neq0$ is the reduction "we may assume $\alpha\neq0$" made in the source, $\alpha$ being a frequency in $\mathbb R/\mathbb Z$.

This is the sharpest of the three bounds collected in Lemma 3.1, and it is what supplies the decay in $\alpha$ on which the circle method of the paper rests. The case $k=1$ is used in Corollary 3.2 and Lemma 3.3; the case $k=2$ is used in the proof of Proposition 4.8, where the quadratic decay $|2\sin\pi\alpha|^{-2}$ is what makes the major-arc $L^{2}$ mass summable.

Formalization Note The bi-infinite series is an unconditional sum over $\mathbb Z$; it converges because compact support leaves only finitely many nonzero terms. Smoothness is infinite continuous differentiability on all of $\mathbb R$, and $F^{(k)}$ is the $k$-fold iterated one-variable derivative.