Lattices with Large Kissing Number from Local Density
Abstract
Let κp(n) denote the maximum kissing number of a lattice ℒ ⊂ ℝn. We prove that κp(n) ≥ 2Ω(√n) for all p≥ 1 and that κp(n) ≥ Ω(αpn) for all p > 2, where αp > 1 is a constant depending only on p. In particular, our former result improves on the previously best known lower bound of κ2(n) ≥ 2Θ(log2 n) for the ℓ2 kissing number, which has stood for over $60$ years. (In concurrent and independent work, Laarhoven and Kominers [arXiv, 2026] showed a similar lower bound on κ2(n) using similar techniques.) In fact, our lower bounds follow from efficient (randomized) constructions of lattices with large kissing number. Our results use a general construction of lattices with large kissing number. Specifically, we show a simple equivalence between such lattices and a new object that we call an exact locally dense lattice. Closely related objects appear throughout the literature on computational lattice problems. Our constructions of lattices with large kissing number then follow as corollaries by showing that constructions due to Bennett and Peikert [RANDOM, 2023] (using Craig lattices) and Aggarwal and Stephens-Davidowitz [STOC, 2018] (using sparsification of ℤn) in fact give exact locally dense lattices, as well as a new construction of exact locally dense lattices built from rotations of the m-dimensional checkerboard lattice Dm.
Related
- Large-Scale Autonomous Discovery of Kissing Number Constructions
- Representations and n-universality of hermitian lattices over local and number fields
- On the kissing number of hyperbolic surfaces with a small systole
- Improved lower bounds for the kissing number in dimensions 49 to 61
- Finiteness theorems for trace Euclidean quadratic lattices over totally real number fields
BibTeX
@misc{bennett2026lattices,
title = {{Lattices with Large Kissing Number from Local Density}},
author = {Huck Bennett and Minki Hhan and Zihan Li and Noah Stephens-Davidowitz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14380},
eprint = {2610.14380},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14380},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lattices with Large Kissing Number from Local Density
AU - Huck Bennett
AU - Minki Hhan
AU - Zihan Li
AU - Noah Stephens-Davidowitz
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14380
UR - https://arxiv.org/abs/2610.14380
AB - Let κ_p(n) denote the maximum kissing number of a lattice ℒ ⊂ ℝ^n. We prove that κ_p(n) ≥ 2^Ω(√n) for all p≥ 1 and that κ_p(n) ≥ Ω(α_p^n) for all p > 2, where α_p > 1 is a constant depending only on p. In particular, our former result improves on the previously best known lower bound of κ_2(n) ≥ 2^Θ(log^2 n) for the ℓ_2 kissing number, which has stood for over $60$ years. (In concurrent and independent work, Laarhoven and Kominers [arXiv, 2026] showed a similar lower bound on κ_2(n) using similar techniques.) In fact, our lower bounds follow from efficient (randomized) constructions of lattices with large kissing number. Our results use a general construction of lattices with large kissing number. Specifically, we show a simple equivalence between such lattices and a new object that we call an exact locally dense lattice. Closely related objects appear throughout the literature on computational lattice problems. Our constructions of lattices with large kissing number then follow as corollaries by showing that constructions due to Bennett and Peikert [RANDOM, 2023] (using Craig lattices) and Aggarwal and Stephens-Davidowitz [STOC, 2018] (using sparsification of ℤ^n) in fact give exact locally dense lattices, as well as a new construction of exact locally dense lattices built from rotations of the m-dimensional checkerboard lattice D_m.
ER -