Equidistant Affine Subspaces: Explicit Bounds, Codimension-Two Reduction, and Transverse Products

Abstract

Let 𝒩(k,n) be the largest size of a family of affine k-planes in Rⁿ at pairwise separation distance one. We develop a unified treatment of upper bounds, dimension reduction, and product constructions. Orthogonal-projection coordinates give a rank-stratified distance identity; a Veronese lifting converts its polynomial consequences into simultaneous total isotropy. Connected Nash decomposition and an analytic diagonal argument replace the linear-independence step in Höfer's three-dimensional method. An explicit cylindrical decomposition estimate yields 𝒩(k,n)leqslant 2^{2^{3(k+1)(n−k)}}. For configurations whose direction sums have dimension at most k+1, we prove the exact reduction 𝒩_{leqslant k+1}(k,n)=𝒩(1,n−k+1). Every codimension-two equidistant configuration splits off a common linear factor, giving 𝒩(n−2,n)=𝒩(1,3). We also determine the sharp bound 2(n−k) when all directions lie in a fixed (k+1)-dimensional subspace. A transverse product construction gives multiplicative inequalities and makes S(c)=𝒩(c−1,2c−1) supermultiplicative. In particular, S(c)geqslant 2^{cbmod 2}7^{⌊ c/2⌋}. The structural results and constructions do not depend on the exact value of 𝒩(1,3).

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BibTeX RIS

BibTeX

@misc{li2026equidistant,
  title = {{Equidistant Affine Subspaces: Explicit Bounds, Codimension-Two Reduction, and Transverse Products}},
  author = {Jia Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37421},
  eprint = {2610.37421},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37421},
}

RIS

TY  - UNPB
TI  - Equidistant Affine Subspaces: Explicit Bounds, Codimension-Two Reduction, and Transverse Products
AU  - Jia Li
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37421
UR  - https://arxiv.org/abs/2610.37421
AB  - Let 𝒩(k,n) be the largest size of a family of affine k-planes in R^n at pairwise separation distance one. We develop a unified treatment of upper bounds, dimension reduction, and product constructions. Orthogonal-projection coordinates give a rank-stratified distance identity; a Veronese lifting converts its polynomial consequences into simultaneous total isotropy. Connected Nash decomposition and an analytic diagonal argument replace the linear-independence step in Höfer's three-dimensional method. An explicit cylindrical decomposition estimate yields 𝒩(k,n)leqslant 2^ 2^ 3(k+1)(n−k). For configurations whose direction sums have dimension at most k+1, we prove the exact reduction 𝒩_leqslant k+1(k,n)=𝒩(1,n−k+1). Every codimension-two equidistant configuration splits off a common linear factor, giving 𝒩(n−2,n)=𝒩(1,3). We also determine the sharp bound 2(n−k) when all directions lie in a fixed (k+1)-dimensional subspace. A transverse product construction gives multiplicative inequalities and makes S(c)=𝒩(c−1,2c−1) supermultiplicative. In particular, S(c)geqslant 2^cbmod 27^⌊ c/2⌋. The structural results and constructions do not depend on the exact value of 𝒩(1,3).
ER  -

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