ON THE DISTRIBUTION OF ARGUMENTS OF GAUSS SUMS

Let F-q be a finite field of q elements of characteristic p. N. M. Katz and Z. Zheng have shown the uniformity of distribution of the arguments arg G(a,chi) of all (q - 1)(q - 2) nontrivial Gauss sums G(a, chi) =Sigma(x is an element of F4)chi(x) exp(2 pi i Tr(ax)/p). where chi is a non-principal Multiplicative character of the multiplicative group F-q* and Tr(z) is the trace of z is an element of F-q into F-p. Here we obtain a similar result for the set of arguments arg G(a,chi) when a and chi run through arbitrary (but sufficiently large) subsets A and X of F-q* and the set of all multiplicative characters of F-q* respectively.