TITLE: Is Weil's bound for Kloosterman sums ever attained?
QUESTION [26 upvotes]: Weil's bound for Kloosterman sums states that for $(a,b)\not=(0,0)$,
$$
|K(a,b;q)|:=\left|\sum_{x\in\mathbb{F}_q^*}\chi(ax+bx^{-1})\right|\leq 2\sqrt{q},
$$
where $\chi$ is a non-trivial additive character on $\mathbb{F}_q$ (the field with $q$ elements).
My question is, is it known to be false that $\sqrt{q}$ can be replaced by $\sqrt{q-1}$?
Here's what is known (to me):

Weil's bound follows from the fact that $K(a,b;q)=\alpha+\beta$ where $\alpha\beta =q$ and $|\alpha|=|\beta|=\sqrt q$.
Thus there is a unique angle $\theta(a,b;q)$ in $[0,\pi]$ such that
$$
\frac{K(a,b;q)}{2\sqrt q}=\cos\theta(a,b;q)
$$
My question then asks, is there $a,b,q$ such that
$$
|\cos\theta(a,b;q)|>\sqrt{1-\frac 1q}?\qquad (*)
$$
"Vertical" equidistribution of Kloosterman angles implies that as $q\to\infty$
$$
\frac 1{q-1}\sum_{\lambda\in F_q^*}f(\theta(1,\lambda;q))\to\frac 2\pi\int_0^\pi f(\theta)\sin^2\theta\,d\theta
$$
Thus for any fixed $\delta>0$, as $q\to\infty$ the proportion of angles $\theta(a,b;q)\in [0,\delta]$ approaches $\frac 1\pi (\delta-\frac 12\sin(2\delta))\approx \frac{2\delta^3}{3\pi}$.
$(*)$ is roughly equivalent to $|\theta(a,b;q)|<q^{-1/2}$, so by equidistribution the expected number of such angles is $\approx 2(q-1)\frac{2}{3\pi} q^{-3/2}\approx \frac{4}{3\pi} q^{-1/2}$, which is (much) less than 1.

So one might ask how good is the concentration around the expected number of angles? And how good is this approximation of the expectation to begin with?
Probably the most reasonable approach is to just search by computer. For $q=p$ prime and $p\leq 61$ there are no counterexamples, but this isn't very convincing.

REPLY [9 votes]: Part of the question asked about how good the concentration of Kloosterman angles is around the expected measure.  It turns out that the discrepancy in the problem of ``vertical distribution" of Kloosterman angles was worked out by Niederreiter: The distribution of values of Kloosterman sums in Arch. der Math. (1991) pages 270--277.  The proof is based on Erdos-Turan type Fourier analysis together with Katz's theorem on equidistribution (essentially what I had in mind in my comment to the question above, but of course carried out 25 years back!).  From Niederreiter's bound on the discrepancy one immediately gets that there are Kloosterman sums $K(1,a;p)$ for each large prime $p$ which are larger than $2\sqrt{p} - C p^{\frac 13}$ for some constant $C$.  One can improve this a little bit and replace $Cp^{\frac 13}$ by $Cp^{\frac 14}$ by working through the argument more carefully in this special situation. (The vanishing of the measure at $0$ makes the general discrepancy bound a little weaker in this situation.)