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On P.23 of https://arxiv.org/pdf/1705.01703.pdf, they seemed to suggest that by the non-negativity of $\psi\big(\frac{k}{N}\big)$ for all $k$, $$ K_N(\xi_0 n)\left[1-\cos\bigg(\frac{2\pi\xi_0 n}{p}\...

Let $v$ be a vector $v \in \mathbb{R}^p$, with non-negative entries and $p$ prime. The Hausdorff-Young inequality gives bounds of the form: $$\|\mathcal{F}v\|_a \le C_{a,b} \|v\|_b$$ where the ...