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 $2$.

翻译：对整系数满秩格$\mathcal{L}\subset\mathbb{R}^n$，Regev与Stephens-Davidowitz证明了\[N_{=k}(\mathcal{L}):=|\{y\in\mathcal{L}:\lVert y\rVert^2=k\}|\le 2\binom{n+2k-2}{2k-1}.\] 本文分类了等号成立情形。当$n\ge2$时，等号成立当且仅当$k=1$且$\mathcal{L}\cong\mathbb{Z}^n$，或$n=8$、$k=2$且$\mathcal{L}\cong E_8$。当$n=1$时，等号成立恰在$\mathcal{L}$表示$k$时。证明表明等号成立具有刚性：壳界的饱和迫使归一化的$k$-范数壳成为对径紧致球面$(4k-1)$-设计。相关的Delsarte--Goethals--Seidel零化多项式导出算术根条件，该条件在$k=2$时孤立出$E_8$格、排除$k=3$情形，并结合Bannai--Damerell/Bannai定理及初等圆论论证，在维数至少为2时排除了所有剩余情形。