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PreprintCC BY 4.0arXiv

Equidistribution of number theoretic point sets in the torus and the sphere in Wasserstein metric and L2 discrepancy

  • Bence Borda
  • Filippo Giannoni
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Abstract

We find the exact rate of equidistribution of several arithmetic point configurations in Wasserstein metric and L2 discrepancy with respect to balls. Kronecker sequences in the torus and the Fibonacci lattice on the Euclidean unit sphere are shown to be more evenly distributed in Wasserstein metric than random points. Modular hyperbolas and modular parabolas in the torus behave like random points, although the constants in the asymptotics of the L2 discrepancy with respect to balls differ from the random case. We also prove a sharp inequality between the quadratic Wasserstein metric and the L2 discrepancy with respect to balls for an arbitrary probability measure on the torus or the sphere. In particular, we show that any point set that is optimally close to the uniform measure in L2 discrepancy with respect to balls is also optimal in the quadratic Wasserstein metric.

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Cite

BibTeXRIS
BibTeX
@misc{borda2026equidistribution,
  title = {{Equidistribution of number theoretic point sets in the torus and the sphere in Wasserstein metric and $L^2$ discrepancy}},
  author = {Bence Borda and Filippo Giannoni},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10881},
  eprint = {2610.10881},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10881},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Equidistribution of number theoretic point sets in the torus and the sphere in Wasserstein metric and L^2 discrepancy
AU  - Bence Borda
AU  - Filippo Giannoni
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10881
UR  - https://arxiv.org/abs/2610.10881
AB  - We find the exact rate of equidistribution of several arithmetic point configurations in Wasserstein metric and L^2 discrepancy with respect to balls. Kronecker sequences in the torus and the Fibonacci lattice on the Euclidean unit sphere are shown to be more evenly distributed in Wasserstein metric than random points. Modular hyperbolas and modular parabolas in the torus behave like random points, although the constants in the asymptotics of the L^2 discrepancy with respect to balls differ from the random case. We also prove a sharp inequality between the quadratic Wasserstein metric and the L^2 discrepancy with respect to balls for an arbitrary probability measure on the torus or the sphere. In particular, we show that any point set that is optimally close to the uniform measure in L^2 discrepancy with respect to balls is also optimal in the quadratic Wasserstein metric.
ER  -