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

Let ${\mu }_q$ be the distribution of the normalized Kloosterman sums $\mathrm{K}\mathrm{l}(a;q)$, $a\in {\mathbb{F}}_q^{\times }$, and let ${\mu }_{\mathrm{S}\mathrm{T}}$ be the Sato–Tate law. We prove $W_s({\mu }_q,{\mu }_{\mathrm{S}\mathrm{T}})\ll q^{-1\mathrm{/}3}$ for $1\le s\le 2$, $W_s({\mu }_q,{\mu }_{\mathrm{S}\mathrm{T}})\ll q^{-1\mathrm{/}6-1\mathrm{/}(3s)}$ for $2\le s<\mathrm{\infty }$, and $W_{\mathrm{\infty }}({\mu }_q,{\mu }_{\mathrm{S}\mathrm{T}})\ll q^{-1\mathrm{/}6}$, answering a question of Kowalski and Untrau. More generally, we bound $W_s(\mu ,{\mu }_{\mathrm{S}\mathrm{T}})$ for all $s$ whenever the Chebyshev moments of $\mu$ satisfy bounds of Katz type. For $s=2$ the bound also follows from a smoothing inequality of Borda and Cuenin; for $s>2$ and $s=\mathrm{\infty }$ 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 ${\mu }_{\mathrm{S}\mathrm{T}}$; and a sharp comparison between $W_s$ and the Kolmogorov distance for laws with square-root edges. The last result shows that $q^{-1\mathrm{/}6}$ is the best rate for $W_{\mathrm{\infty }}$ obtainable from Niederreiter’s discrepancy bound. Numerically, for primes ${10}^3<p<{10}^6$, all these distances appear to decay like $p^{-1\mathrm{/}2}$.