Let be a finite field of q elements. For multiplicative characters of , we let denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for , the normalized Jacobi sum ( nontrivial) is asymptotically equidistributed on the unit circle as , when and run through all nontrivial multiplicative characters of . In this paper, we show a similar property for . More generally, we show that the normalized Jacobi sum ( nontrivial) is asymptotically equidistributed on the unit circle, when run through arbitrary sets of nontrivial multiplicative characters of with two of the sets being sufficiently large. The case answers a question of Shparlinski.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11371043
Award Identifier / Grant number: 11321101
Funding statement: The first-named author was partially supported by the National Natural Science Foundation of China under grant 11371043. The second-named author was partially supported by China’s Recruitment Program of Global Experts, the National Natural Science Foundation of China under grant 11321101, the Hua Loo-Keng Key Laboratory of Mathematics (Chinese Academy of Sciences) and the National Center for Mathematics and Interdisciplinary Sciences (Chinese Academy of Sciences). The third-named author was partially supported by Program 863 grant 2013AA013702 and Program 973 grant 2013CB834205.
The authors wish to thank Ming Fang, Ofer Gabber, and Nicholas Katz for useful discussions, as well as the referee for helpful comments. The first- and second-named author gratefully acknowledge the hospitality and support of l’Institut des Hautes Études Scientifiques, where part of this work was done during their visits.
 P. Deligne, Application de la formule des traces aux sommes trigonométriques, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 41/2), Lecture Notes in Math. 569, Springer, Berlin (1977), 168–232. 10.1007/BFb0091523Search in Google Scholar
 P. Erdös and P. Turán, On a problem in the theory of uniform distribution. I, II, Nederl. Akad. Wetensch. Proc. 51 (1948), 1146–1154, 1262–1269. Search in Google Scholar
 A. Grothendieck, Formule d’Euler–Poincaré en cohomologie étale, Séminaire de Géometrie Algébrique du Bois-Marie 1965–1966 (SGA 5), Lecture Notes in Math. 589, Springer, Berlin (1977), 372–406. 10.1007/BFb0096810Search in Google Scholar
 N. M. Katz and Z. Zheng, On the uniform distribution of Gauss sums and Jacobi sums, Analytic number theory, Vol. 2 (Allerton Park 1995), Progr. Math. 139, Birkhäuser, Boston (1996), 537–558. Search in Google Scholar
 R. C. King, Modification rules and products of irreducible representations of the unitary, orthogonal, and symplectic groups, J. Math. Phys. 12 (1971), 1588–1598. 10.1063/1.1665778Search in Google Scholar
 F. E. Su, A LeVeque-type lower bound for discrepancy, Monte Carlo and quasi-Monte Carlo methods (Claremont 1998), Springer, Berlin (2000), 448–458. 10.1007/978-3-642-59657-5_31Search in Google Scholar
 H. Weyl, The classical groups, Princeton Landmarks in Math., Princeton University Press, Princeton 1997. Search in Google Scholar
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