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, χ) of all (q - 1)(q - 2) nontrivial Gauss sums. Where χ is a non-principal multiplicative character of the multiplicative group F_q* and Tr(z) is the trace of z ∈ F_q into F_p. Here we obtain a similar result for the set of arguments arg G(a, χ) when a and χ run through arbitrary (but sufficiently large) subsets A and X of F_q* and the set of all multiplicative characters of F_q*, respectively.