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PreprintCC BY 4.0arXiv

Counterexamples to the Ramos conjecture for two hyperplanes

  • Florian Frick
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Abstract

For every n≥2, we construct 4n−2 nondegenerate Gaussian measures on ℝ6n−3 that cannot be simultaneously equipartitioned by two affine hyperplanes. This disproves the Ramos conjecture for two hyperplanes. Combined with known upper bounds, the construction shows that 3·2s−1−2 is the least dimension guaranteeing a common two-hyperplane equipartition of 2s−2 absolutely continuous probability measures, for every s≥3. We characterize the Gaussian equipartition threshold in terms of the least number of positive definite quadratic measurements needed for phase retrieval. Modified complex polynomial multiplication gives 2r−2 positive definite measurements in every even dimension r≥4. This number is optimal when r=2k+2, k≥1.

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Cite

BibTeXRIS
BibTeX
@misc{frick2026counterexamples,
  title = {{Counterexamples to the Ramos conjecture for two hyperplanes}},
  author = {Florian Frick},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26723},
  eprint = {2609.26723},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26723},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Counterexamples to the Ramos conjecture for two hyperplanes
AU  - Florian Frick
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26723
UR  - https://arxiv.org/abs/2609.26723
AB  - For every n≥2, we construct 4n−2 nondegenerate Gaussian measures on ℝ^6n−3 that cannot be simultaneously equipartitioned by two affine hyperplanes. This disproves the Ramos conjecture for two hyperplanes. Combined with known upper bounds, the construction shows that 3·2^s−1−2 is the least dimension guaranteeing a common two-hyperplane equipartition of 2^s−2 absolutely continuous probability measures, for every s≥3. We characterize the Gaussian equipartition threshold in terms of the least number of positive definite quadratic measurements needed for phase retrieval. Modified complex polynomial multiplication gives 2r−2 positive definite measurements in every even dimension r≥4. This number is optimal when r=2^k+2, k≥1.
ER  -