On the distribution of multivariate Jacobi sums

TL;DR

This paper proves that normalized multivariate Jacobi sums become uniformly distributed on the unit circle as the size of the character sets grows under certain conditions, extending previous distribution results.

Contribution

It extends the known distribution results of Jacobi sums to more general sets of characters and broader conditions on their sizes.

Findings

Normalized Jacobi sums are asymptotically equidistributed on the unit circle.

Distribution holds for arbitrary sets of nontrivial characters with large enough size.

Results generalize previous work by Xi, Zheng, and the authors.

Abstract

Let $\mathbf{F}_q$ be a finite field of $q$ elements. We show that the normalized Jacobi sum $q^{-(m-1)/2}J(\chi_1,\dots,\chi_m)$ ($\chi_1\dotsm \chi_m$ nontrivial) is asymptotically equidistributed on the unit circle, when $\chi_1\in \mathcal{A}_1,\dots, \chi_m\in \mathcal{A}_m$ run through arbitrary sets of nontrivial multiplicative characters of $\mathbf{F}_q^\times$, if $\#\mathcal{A}_1\ge q^{\frac{1}{2}+\epsilon}$, $\#\mathcal{A}_2 \ge (\log q)^{\frac{1}{\delta}-1}$ for $\epsilon>\delta>0$ fixed and $q\to \infty$ or if $\#\mathcal{A}_1\#\mathcal{A}_2/q\to \infty$. This extends previous results of Xi, Z. Zheng, and the authors.