Wasserstein rates of all orders for the Sato-Tate law of Kloosterman sums

Source abstract

Let $μ_q$ be the distribution of the normalized Kloosterman sums $\mathrm{Kl}(a;q)$, $a\in\mathbb{F}_q^\times$, and let $μ_{\mathrm{ST}}$ be the Sato--Tate law. We prove $W_s(μ_q,μ_{\mathrm{ST}})\ll q^{-1/3}$ for $1\le s\le2$, $W_s(μ_q,μ_{\mathrm{ST}})\ll q^{-1/6-1/(3s)}$ for $2\le s 2$ and $s=\infty$ our bounds are stronger. The proof uses the quantile coupling, which arises from Stieltjes sums over two different intervals whose cells are paired so as to balance two measures, and which is optimal for all $s$ simultaneously; an explicit $W_2$ smoothing inequality relative to $μ_{\mathrm{ST}}$; and a sharp comparison between $W_s$ and the Kolmogorov distance for laws with square-root edges. The last result shows that $q^{-1/6}$ is the best rate for $W_\infty$ obtainable from Niederreiter's discrepancy bound. Numerically, for primes $10^3<p<10^6$, all these distances appear to decay like $p^{-1/2}$.