Equilateral sets in ℓ_pⁿ beyond n+1: Linear excess and dimension cost

@misc{chalmers2026equilateral,
  title = {{Equilateral sets in $\ell_p^n$ beyond $n+1$: Linear excess and dimension cost}},
  author = {Logan R. Chalmers and Aran S. Ziegler},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13983},
  eprint = {2610.13983},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13983},
}

TY  - UNPB
TI  - Equilateral sets in ℓ_p^n beyond n+1: Linear excess and dimension cost
AU  - Logan R. Chalmers
AU  - Aran S. Ziegler
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.13983
UR  - https://arxiv.org/abs/2610.13983
AB  - How many equilateral points can ℓ_p^n contain beyond n+1? For every fixed real p>4, we construct equilateral sets for which the number of points beyond n+1 grows at least linearly with n. For every fixed dimension, we prove that the maximum is n+1 at all sufficiently large finite p, with an explicit threshold. The least dimension admitting any fixed number of extra points is p^1+o(1) as p→∞. The constructions and finite bounds are formalised in Lean. Thus p=4 marks the boundary: n+1 is exact in every dimension for 2≤ p≤4, while at every fixed finite p>4 the excess grows at least linearly with dimension.
ER  -