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Covering Projective Height Balls by Subspaces in Rigid Adelic Spaces

  • Ruida Di
  • Runjie Hu
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Abstract

Let E be an n-dimensional rigid adelic space over a number field K. We study the minimum number gE(R) of proper K-subspaces needed to cover the projective height ball of radius R, together with the maximum cardinality hE(R) of a subset in linear general position. We show that, once R is sufficiently large compared with the last Roy--Thunder minimum of E, both quantities have order ΨE(R)[K:ℚ], where ΨE(R) is an explicit expression in the Roy--Thunder minima. The comparison constants are effectively computable and uniform in E. For the standard adelic space Kn, this gives order R[K:ℚ]n/(n−1).

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Cite

BibTeXRIS
BibTeX
@misc{di2026covering,
  title = {{Covering Projective Height Balls by Subspaces in Rigid Adelic Spaces}},
  author = {Ruida Di and Runjie Hu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00988},
  eprint = {2609.00988},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00988},
}
RIS
TY  - UNPB
TI  - Covering Projective Height Balls by Subspaces in Rigid Adelic Spaces
AU  - Ruida Di
AU  - Runjie Hu
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00988
UR  - https://arxiv.org/abs/2609.00988
AB  - Let E be an n-dimensional rigid adelic space over a number field K. We study the minimum number g_E(R) of proper K-subspaces needed to cover the projective height ball of radius R, together with the maximum cardinality h_E(R) of a subset in linear general position. We show that, once R is sufficiently large compared with the last Roy--Thunder minimum of E, both quantities have order Ψ_E(R)^[K:ℚ], where Ψ_E(R) is an explicit expression in the Roy--Thunder minima. The comparison constants are effectively computable and uniform in E. For the standard adelic space K^n, this gives order R^[K:ℚ]n/(n−1).
ER  -