Kurlberg–Rudnick conjecture for normalized exponential sums

For a finite field $\mathbb{F}_{q}$, let $\mathcal{C}$ be the multiplicative group $\mathbb{F}_{q}^{*}$ or the group of norm-one elements in a quadratic extension, and let $E_q(\nu,\chi)$ be the normalized exponential sum defined by

$$E_q(\nu,\chi)=\frac{1}{|\mathcal{C}|}\sum_{1\neq x\in\mathcal{C}} e_q\left(\nu\kappa\frac{x+1}{x-1}\right)\chi(x)\chi_2(x),$$

where $\chi$ ranges over characters of $\mathcal{C}$, $\chi_2$ is its quadratic character, and $\nu\in\mathbb{F}_q$ and $\kappa$ are as in the construction. Kurlberg–Rudnick's conjecture. For fixed $0\neq\nu\in\mathbb{F}_{q}$, the points $\sqrt{q}E_q(\nu,\chi)$ become equidistributed on $[-2,2]$ with respect to the Sato–Tate measure as $q\to\infty$. For distinct $\nu_1,\ldots,\nu_r\in\mathbb{F}_q$, the corresponding limiting distributions are those of $r$ independent random variables. This conjecture predicts the limiting behavior underlying the quantum unique ergodicity distributions for symplectic linear maps; its multidimensional consequences are developed later in the paper, but the conjecture itself is not resolved here.