On the distribution of arguments of Gauss sums
Let \(p\) be a prime, \(q\) a \(p\)-power, \(\mathbb F_q\) the field of order \(q\) and \(\text{Tr}\) the trace from \(\mathbb F_q\) to \(\mathbb F_p\). For \(a\in \mathbb F_q^*\) and a non-principal character \(\chi\) on \(\mathbb F_q^*\), the Gauss sum is: \[ G(a, \chi )=\sum_{x\in \mathbb F_q^*} \chi (x)\,\text{exp}(2\pi i\,\text{Tr}(ax)/p). \] The argument \(\operatorname{arg}G(a, \chi )\) is defined by \(G(a, \chi )=\text{exp}(i\,\operatorname{arg} G(a, \chi ))q^{1/2}\). \textit{N. M. Katz} and \textit{Z. Zheng} [Prog. Math. 139, 537--558 (1996; Zbl 0866.11073)] have shown that if \(a\) runs over all of \(\mathbb F_q^*\) and \(\chi\) runs over all non-principal characters then \(\operatorname{arg} G(a, \chi )/2\pi \) is asymptotically uniformly distributed in \([0,1]\). Here a similar result is shown if \(a\) runs over a subset \(A\) of \(\mathbb F_q^*\) and \(\chi\) runs over a subset \(X\) of non-principal characters, provided \(|A|\) and \(|X|\) are sufficiently large. Specifically, it suffices to have \(|A|\,|X|\geq q^{1+\varepsilon}\) for fixed \(\varepsilon >0\) and \(q\) large enough. Conversely, the author constructs subsets \(A\) and \(X\) with \(|A|\,|X|=(q-1)/2\) for which \(\text{arg}\,G(a, \chi )\) is constant on \(A\times X\).
scientific article; zbMATH DE number 5554653