Equality in a Reverse Minkowski Shell Bound for Integral Lattices via Spherical Designs

2026-05-24

AI Summary

This work completely characterizes all cases in which the upper bound on the number of shells of an integral lattice, as proposed by Regev and Stephens-Davidowitz, is attained. By normalizing shells to spherical designs and systematically applying the Delsarte–Goethals–Seidel annihilating polynomials, the Bannai–Damerell theorem, and arithmetic root conditions, the authors classify the lattice structures achieving equality. The main contribution establishes that for dimension \(n \geq 2\), equality holds if and only if either \(k=1\) and the lattice is isomorphic to \(\mathbb{Z}^n\), or \(n=8\), \(k=2\), and the lattice is isomorphic to \(E_8\); for \(n=1\), equality occurs precisely when the lattice represents \(k\). These results uncover a deep connection between this extremal problem and tight antipodal spherical designs, ruling out all other lattices in higher dimensions.

Abstract

For a full-rank integral lattice $\mathcal{L}\subset\mathbb{R}^n$, Regev and Stephens-Davidowitz proved that \[N_{=k}(\mathcal{L}):=|\{y\in\mathcal{L}:\lVert y\rVert^2=k\}|\le 2\binom{n+2k-2}{2k-1}.\] We classify the equality cases. For $n\ge2$, equality holds if and only if either $k=1$ and $\mathcal{L}\cong\mathbb{Z}^n$, or $n=8$, $k=2$, and $\mathcal{L}\cong E_8$. For $n=1$, equality holds exactly when $\mathcal{L}$ represents $k$. The proof shows that equality is rigid. Saturation of the shell bound forces the normalized norm-$k$ shell to be an antipodal tight spherical $(4k-1)$-design. The associated Delsarte--Goethals--Seidel annihilator polynomial gives an arithmetic root condition, which isolates $E_8$ at $k=2$, rules out $k=3$, and combines with the Bannai--Damerell/Bannai theorem and an elementary circle argument to exclude all remaining cases in dimension at least $2$.

Problem

• integral lattices
• spherical designs
• Minkowski shell bound
• tight designs
• equality cases

Innovation

• spherical designs
• integral lattices
• Minkowski shell bound
• tight designs
• E8 lattice