The distribution of values of Kloosterman sums (Q1174731)

Let \(F_ q\) be the finite field of order \(q\), and for a fixed additive character \(\chi\) of \(F_ q\) let \(K(q,a)=\sum_{b\in F_ q^*}\chi(b+ab^{-1})\) be the Kloosterman sum, which, as a consequence of \(A\). Weil's upper bound satisfies \(K(q,a)=2q^{1/2}w(q,a)\) with \(w(q,a)\in[-1,1]\). From the results of \textit{N. M. Katz} [Gauss sums, Kloosterman sums and monodromy groups (Ann. Math. Stud. 116) (Princeton, 1988; Zbl 0675.14004)] one knows, that, as \(q\to\infty\) the \(q-1\) numbers \(w(q,a)\), \(a\in F_ q^*\), have an asymptotic distribution given by the measure \((2/\pi)(1-t^ 2)^{1/2}dt\) on \([-1,1]\). In the paper under review a more refined result is proved. For this let for \(x\in[-1,1]\), \[ G(x)=(2/\pi)\int_{-1}^ x (1-t^ 2)^{1/2}dt,\qquad A([- 1,x);q)=\text{card}\{a\in F_ q^*:\;-1\leq w(q,a)<x\}, \] \[ R_ q(x)=A([-1,x);q)/q-G(x),\qquad D^*_ q=\sup_{x\in[-1,1]}| R_ q(x)|. \] The main result of the article then reads \(D_ q^*<10q^{-1/4}\).