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Bounds for lattice points in terms of volume and successive minima

  • Jun Wang
  • Fei Xue
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Abstract

We prove, in every dimension, the upper bound conjectured by Betke, Henk and Wills for the lattice-point enumerator in terms of volume and successive minima. For every convex body K⊂ Rn and full-rank lattice Λ, we establish #(K∩Λ)≤voln(K)/detΛ∏i=1n(1+iλi/2), where λi are the successive minima of (K−K)/2 with respect to Λ. Equality holds precisely for lattice translates of positive integral dilates of unimodular simplices. We also prove #(intK∩Λ)≥voln(K)/detΛ (1−n/4∑i=1n−1λi−n/2λn), which implies the lower estimate with factor 1−1/2∑i=1n iλi. In dimension two we prove the lower bound with factor (1−λ1/2)(1−λ2) when λ2≤1. We also observe that a conjectured product lower bound for origin-symmetric bodies fails in dimension three. The proofs combine geometric compression, lattice-cell counts in coordinate sections, and an elementary coefficient comparison.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026bounds,
  title = {{Bounds for lattice points in terms of volume and successive minima}},
  author = {Jun Wang and Fei Xue},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41125},
  eprint = {2610.41125},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41125},
}
RIS
TY  - UNPB
TI  - Bounds for lattice points in terms of volume and successive minima
AU  - Jun Wang
AU  - Fei Xue
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41125
UR  - https://arxiv.org/abs/2610.41125
AB  - We prove, in every dimension, the upper bound conjectured by Betke, Henk and Wills for the lattice-point enumerator in terms of volume and successive minima. For every convex body K⊂ R^n and full-rank lattice Λ, we establish #(K∩Λ)≤vol_n(K)/detΛ∏_i=1^n(1+iλ_i/2), where λ_i are the successive minima of (K−K)/2 with respect to Λ. Equality holds precisely for lattice translates of positive integral dilates of unimodular simplices. We also prove #(intK∩Λ)≥vol_n(K)/detΛ (1−n/4∑_i=1^n−1λ_i−n/2λ_n), which implies the lower estimate with factor 1−1/2∑_i=1^n iλ_i. In dimension two we prove the lower bound with factor (1−λ_1/2)(1−λ_2) when λ_2≤1. We also observe that a conjectured product lower bound for origin-symmetric bodies fails in dimension three. The proofs combine geometric compression, lattice-cell counts in coordinate sections, and an elementary coefficient comparison.
ER  -