Wasserstein rates of all orders for the Sato-Tate law of Kloosterman sums
Abstract
We answer, in the trace metric, a question of Kowalski and Untrau on Wasserstein distances of all orders between the distribution μq of the normalized Kloosterman sums Kl(a;q), a∈𝔽q×, and the Sato-Tate law μST: we prove bounds for every order s and show that their exponents are optimal for Katz-type moment bounds. In the trace metric we prove Ws(μq,μST)≪s q−1/3 for 1≤ s<3, W3≪ q−1/3(log q)1/3, Ws≪s q−1/4−1/(4s) for 3<s<∞ and W∞≪ q−1/4. In the angle metric we obtain bounds of every order, including W2≤ 7q−1/3 and W∞≪ q−1/6. These estimates follow from a general theorem for probability measures satisfying Katz-type Chebyshev moment bounds. Its proof combines an edge-sensitive discrepancy estimate with a new level-set bound for the distribution function and a quantile-displacement argument. We construct measures showing that the resulting exponents cannot in general be improved using moment bounds of this type alone, in the trace metric and in the angle metric for s≤ 2 and s≥ 6 (up to logarithms); these constructions do not assert optimality for the Kloosterman measures themselves. A sharp comparison between Wasserstein distances and the Kolmogorov distance for the Sato-Tate law exhibits the same transition at s=3. Finally, using the exact third moment, we prove W1(μq,μST)>q−1/2/10 when 3nmid q.
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BibTeX
@misc{rahimi2026wasserstein,
title = {{Wasserstein rates of all orders for the Sato-Tate law of Kloosterman sums}},
author = {Amir M. Rahimi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16169},
eprint = {2610.16169},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16169},
}
RIS
TY - UNPB
TI - Wasserstein rates of all orders for the Sato-Tate law of Kloosterman sums
AU - Amir M. Rahimi
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16169
UR - https://arxiv.org/abs/2610.16169
AB - We answer, in the trace metric, a question of Kowalski and Untrau on Wasserstein distances of all orders between the distribution μ_q of the normalized Kloosterman sums Kl(a;q), a∈𝔽_q^×, and the Sato-Tate law μ_ST: we prove bounds for every order s and show that their exponents are optimal for Katz-type moment bounds. In the trace metric we prove W_s(μ_q,μ_ST)≪_s q^−1/3 for 1≤ s<3, W_3≪ q^−1/3(log q)^1/3, W_s≪_s q^−1/4−1/(4s) for 3<s<∞ and W_∞≪ q^−1/4. In the angle metric we obtain bounds of every order, including W_2≤ 7q^−1/3 and W_∞≪ q^−1/6. These estimates follow from a general theorem for probability measures satisfying Katz-type Chebyshev moment bounds. Its proof combines an edge-sensitive discrepancy estimate with a new level-set bound for the distribution function and a quantile-displacement argument. We construct measures showing that the resulting exponents cannot in general be improved using moment bounds of this type alone, in the trace metric and in the angle metric for s≤ 2 and s≥ 6 (up to logarithms); these constructions do not assert optimality for the Kloosterman measures themselves. A sharp comparison between Wasserstein distances and the Kolmogorov distance for the Sato-Tate law exhibits the same transition at s=3. Finally, using the exact third moment, we prove W_1(μ_q,μ_ST)>q^−1/2/10 when 3nmid q.
ER -