On the distribution of arguments of Gauss sums

Abstract

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

$$G(a, \chi) = \sum_{x \in {\mathbf F}_q} \chi(x) \exp(2 \pi i \mathrm{Tr}(ax)/p),$$

where χ is a non-principal multiplicative character of the multiplicative group F_q^* and Tr(z) is the trace of z $\in$ 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 ${\mathscr A}$ and ${\mathscr X}$ of F_q^* and the set of all multiplicative characters of F_q^*, respectively.