Discreteness, finite rank, and stability of p-frame energy minimizers

Abstract

We study probability measures minimizing the energy with kernel |x· y|^p on the real unit sphere. For every positive rational p=a/b, each minimizer admits a finitely supported minimizing replacement. If a/b is reduced and b≥ d, every minimizer on 𝕊^d−1 is finitely supported. The latter assertion follows from a finite-rank algebraic lift, Nash curve selection, and a Wronskian multiplicity bound. For irrational exponents, we prove that a positive semidefinite restriction of finite rank cannot have infinite compact support. We also establish local stability and a quantitative support bound under positive-definite Hessians at every contact of every limiting minimizer. This hypothesis is stronger than finite support and does not prove unconditional openness of the discreteness regime. Finally, subtracting a single suitably scaled high even power produces a uniformly nearby kernel whose every minimizer has finite support. We identify the atomic-purity selection of these approximants and prove a target-centered recovery theorem, separating finite approximation from discreteness of the limiting problem.

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BibTeX RIS

BibTeX

@misc{park2026discreteness,
  title = {{Discreteness, finite rank, and stability of p-frame energy minimizers}},
  author = {Josiah Park},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32973},
  eprint = {2609.32973},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32973},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - Discreteness, finite rank, and stability of p-frame energy minimizers
AU  - Josiah Park
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32973
UR  - https://arxiv.org/abs/2609.32973
AB  - We study probability measures minimizing the energy with kernel |x· y|^p on the real unit sphere. For every positive rational p=a/b, each minimizer admits a finitely supported minimizing replacement. If a/b is reduced and b≥ d, every minimizer on 𝕊^d−1 is finitely supported. The latter assertion follows from a finite-rank algebraic lift, Nash curve selection, and a Wronskian multiplicity bound. For irrational exponents, we prove that a positive semidefinite restriction of finite rank cannot have infinite compact support. We also establish local stability and a quantitative support bound under positive-definite Hessians at every contact of every limiting minimizer. This hypothesis is stronger than finite support and does not prove unconditional openness of the discreteness regime. Finally, subtracting a single suitably scaled high even power produces a uniformly nearby kernel whose every minimizer has finite support. We identify the atomic-purity selection of these approximants and prove a target-centered recovery theorem, separating finite approximation from discreteness of the limiting problem.
ER  -

Abstract

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