We explore numerically the eigenvalues of the hermitian form$$\sum _{q\le Q}\sum _{amod{}^{*}q}{\left|\sum _{n\le N}{\varphi }_{n}e(na/q)\right|}^2$$when $N={\sum }_{q\le Q}\phi \left(q\right)$. We improve on the existing upper bound, and produce a (conjectural) plot of the asymptotic distribution of its eigenvalues by exploiting fairly extensive computations. The main outcome is that this asymptotic density most probably exists but is not continuous with respect to the Lebesgue measure.

A sum of exponentials of the form $f\left(x\right)=exp\left(2\pi i{N}_{1}x\right)+exp\left(2\pi i{N}_{2}x\right)+\cdots +exp\left(2\pi i{N}_{m}x\right)$, where the $N_k$ are distinct integers is called an idempotent trigonometric polynomial (because the convolution of $f$ with itself is $f$) or, simply, an idempotent. We show that for every $p\>1,$ and every set $E$ of the torus $𝕋=ℝ/\mathbb{Z}$ with $\left|E\right|\>0,$ there are idempotents concentrated on $E$ in the $L^p$ sense. More precisely, for each $p\>1,$ there is an explicitly calculated constant $C_p\>0$ so that for each $E$ with $\left|E\right|\>0$ and $ϵ\>0$ one can find an idempotent $f$ such that the ratio ${\left({\int }_E{\left|f\right|}^p/{\int }_{𝕋}{\left|f\right|}^p\right)}^{1/p}$ is greater than $C_p-ϵ$. This is in fact...