Hello , noting $e(x)$ for $e^{2i\pi x}$,

$E(R):=\{f\vert f(x)=\sum_{0\leq r\leq R-1}a_{r}e(rx)\:,\,\sum_{r}\vert a_{r}\vert^{2}=1\:,\: a_{r}\in\mathbb{C}\:,\:\forall r\}$

$h(a,R):=\inf_{f\in E(R)}\intop_{0}^{a}\vert f(x)\vert^{2}dx$ with $0\leq a\leq 1$ .

I am curious to know the behavior of $h(a,R)$ particularly if $R\rightarrow\infty$

Thank you in advance for an idea !