Equidistributions of Jacobi sums

TL;DR

This paper proves that normalized Jacobi sums become uniformly distributed on the unit circle as the field size grows, with new results on their distribution over specific character subsets, improving previous bounds.

Contribution

It establishes new equidistribution results for Jacobi sums when fixing one character and varying the other, refining earlier work by relaxing size conditions on character subsets.

Findings

Normalized Jacobi sums are equidistributed on the unit circle as q→∞.

New bounds are provided for the distribution over subsets of characters.

Improves previous results requiring larger subsets for equidistribution.

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.