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PreprintCC BY 4.0arXiv

The curse of dimensionality for geometric L1-discrepancies with nonnegative weights

  • Josef Dick
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Abstract

We prove that the normalized star, extreme, and periodic L1-discrepancies with arbitrary nonnegative weights suffer from the curse of dimensionality. A single elementary lemma is used in all three cases: after normalizing the target and every one-dimensional kernel to probability densities, a uniform square-root-affinity gap tensorizes over the coordinates. For every d∈ ℕ and 0<ε<1, the normalized information complexities have lower bounds of the form ((1−ε)2)/(1+ε) qd, with qstar=1/2+1/√3, qext=(75(9+4√2))/784, qper=25/16. All three constants are larger than one. The argument uses only nonnegativity, product integration, the elementary inequalities for the square root, and Cauchy--Schwarz.

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Cite

BibTeXRIS
BibTeX
@misc{dick2026curse,
  title = {{The curse of dimensionality for geometric $L_1$-discrepancies with nonnegative weights}},
  author = {Josef Dick},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.24290},
  eprint = {2607.24290},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.24290},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The curse of dimensionality for geometric L_1-discrepancies with nonnegative weights
AU  - Josef Dick
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2607.24290
UR  - https://arxiv.org/abs/2607.24290
AB  - We prove that the normalized star, extreme, and periodic L_1-discrepancies with arbitrary nonnegative weights suffer from the curse of dimensionality. A single elementary lemma is used in all three cases: after normalizing the target and every one-dimensional kernel to probability densities, a uniform square-root-affinity gap tensorizes over the coordinates. For every d∈ ℕ and 0<ε<1, the normalized information complexities have lower bounds of the form ((1−ε)^2)/(1+ε) q^d, with q_star=1/2+1/√3, q_ext=(75(9+4√2))/784, q_per=25/16. All three constants are larger than one. The argument uses only nonnegativity, product integration, the elementary inequalities for the square root, and Cauchy--Schwarz.
ER  -