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A set of points with optimal L2 spherical cap discrepancy
Abstract
We introduce an explicit deterministic collection of N spherical points, cPN⊂𝕊2, that we call the {\em deterministic Diamond points}. For every 0<α<2 we prove that there exists a constant Cα>0 such that 0≤ (2α+1)/(α+2)N2−∑x,y∈cPN|x−y|α≤ CαN1−α/2. This result shows that cPN⊂𝕊2 has a Riesz energy deficit of optimal order in this range of the parameter. In particular, for α=1, Stolarsky's invariance principle implies that cPN⊂𝕊2 has L2 spherical cap discrepancy of asymptotically optimal order DL2C(𝒫N)asymp N−3/4, making it, to our knowledge, the first explicit configuration proved to have this property. More generally, our result implies that the point set cPN has Sobolev Hs(Sph) worst-case error of optimal order for all 1<s<2.
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BibTeX
@misc{beltran2026points,
title = {{A set of points with optimal $L^2$ spherical cap discrepancy}},
author = {Carlos Beltrán and Jordi Marzo and Joaquim Ortega-Cerdà},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27708},
eprint = {2609.27708},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27708},
}
RIS
TY - UNPB
TI - A set of points with optimal L^2 spherical cap discrepancy
AU - Carlos Beltrán
AU - Jordi Marzo
AU - Joaquim Ortega-Cerdà
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27708
UR - https://arxiv.org/abs/2609.27708
AB - We introduce an explicit deterministic collection of N spherical points, cP_N⊂𝕊^2, that we call the {\em deterministic Diamond points}. For every 0<α<2 we prove that there exists a constant C_α>0 such that 0≤ (2^α+1)/(α+2)N^2−∑_x,y∈cP_N|x−y|^α≤ C_αN^1−α/2. This result shows that cP_N⊂𝕊^2 has a Riesz energy deficit of optimal order in this range of the parameter. In particular, for α=1, Stolarsky's invariance principle implies that cP_N⊂𝕊^2 has L^2 spherical cap discrepancy of asymptotically optimal order D_L^2^C(𝒫_N)asymp N^−3/4, making it, to our knowledge, the first explicit configuration proved to have this property. More generally, our result implies that the point set cP_N has Sobolev H^s(Sph) worst-case error of optimal order for all 1<s<2.
ER -