Equidistributions of Jacobi sums

Abstract

Let $\mathbf{F}_q$ be a finite field of $q$ elements. We show that the normalized Jacobi sum $J(\chi,\eta)/\sqrt{q}$, for each fixed non-trivial multiplicative character $\eta$, becomes equidistributed in the unit circle as $q\rightarrow+\infty,$ when $\chi$ runs over all non-trivial multiplicative characters different from $\eta^{-1}.$ Previously, the similar equidistribution was obtained by Katz and Zheng by varying both of $\chi$ and $\eta$. On the other hand, we also obtain the equidistribution of $J(\chi,\eta)$ as $(\chi,\eta)$ runs over $\mathcal{X}\times\mathcal{Y}\subseteq(\widehat{\mathrm{F}^*})^2$, as long as $|\mathcal{X}|>q^{\frac{1}{2}+\varepsilon}$ and $|\mathcal{Y}|>q^\varepsilon$ for any $\varepsilon>0$. This updates a recent work of Lu, Zheng and Zheng, who require $|\mathcal{X}||\mathcal{Y}|>q\log^2q.$ The main ingredient is the estimate for hypergeometric sums due to Katz.