AI 中文总结
本文建立平坦环面的谱、等周参数与格基本域协方差的关系，得到谱隙和Cheeger常数的下界，部分解决Kannan–Lovász–Simonovits猜想，特定条件下不等式等价。
AI 中文摘要
我们记录了平坦环面$\boldsymbol{\textsf{T}}_\boldsymbol{\boldsymbol{\textsf{\textbf{Λ}}}} = \boldsymbol{\textsf{R}}^n/\boldsymbol{\textsf{Λ}}$的谱参数与等周参数之间，以及格$\boldsymbol{\textsf{Λ}} \boldsymbol{\boldsymbol{\textsf{\textbf{⊂}}}} \boldsymbol{\textsf{R}}^n$的基本域$\boldsymbol{\textsf{K}}$的协方差结构之间的若干初等关系。对于每个可测基本域$\boldsymbol{\textsf{K}}$和对偶格$\boldsymbol{\textsf{Λ}}^*$中的非零向量$\boldsymbol{\textsf{ξ}}$，我们得到了尖锐的方向方差估计：$\boldsymbol{\boldsymbol{\textsf{\textbf{⟨}}}} \boldsymbol{\textsf{Cov}}_\boldsymbol{\textsf{K}} \boldsymbol{\textsf{ξ}},\boldsymbol{\textsf{ξ}} \boldsymbol{\boldsymbol{\textsf{\textbf{⟩}}}} \boldsymbol{\boldsymbol{\textsf{\textbf{≥}}}} \boldsymbol{\boldsymbol{\textsf{\textbf{1/12}}}}$。这给出了环面谱隙$\boldsymbol{\textsf{λ}}_{\boldsymbol{\textsf{SG}}}(\boldsymbol{\textsf{T}}_\boldsymbol{\textsf{Λ}})$（等价于最短非零对偶向量长度$\boldsymbol{\textsf{λ}}_1(\boldsymbol{\textsf{Λ}}^*)$）关于$\boldsymbol{\textsf{K}}$的最大协方差的下界：$\boldsymbol{\textsf{λ}}_{\boldsymbol{\textsf{SG}}}(\boldsymbol{\textsf{T}}_\boldsymbol{\textsf{Λ}}) = \boldsymbol{\boldsymbol{\textsf{\textbf{4π²}}}} \boldsymbol{\textsf{λ}}_1(\boldsymbol{\textsf{Λ}}^*)^2 \boldsymbol{\boldsymbol{\textsf{\textbf{≥}}}} \boldsymbol{\boldsymbol{\textsf{\textbf{π²/(3∥Cov}}}}_\boldsymbol{\textsf{K}} \boldsymbol{\boldsymbol{\textsf{\textbf{∥}}}}_{\boldsymbol{\textsf{op}}}\boldsymbol{\boldsymbol{\textsf{\textbf{)}}}}$。利用Hadwiger的经典论证，我们对等周剖面和Cheeger常数$\boldsymbol{\textsf{D}}_{\boldsymbol{\textsf{Che}}}(\boldsymbol{\textsf{T}}_\boldsymbol{\textsf{Λ}})$也得到了类似的尖锐结果。特别地，当行列式$\boldsymbol{\textsf{det}} \boldsymbol{\textsf{Λ}} = 1$的格的Voronoi胞腔$\boldsymbol{\textsf{K}}_\boldsymbol{\textsf{Λ}}$是各向同性时，Klartag和Lehec对切片问题的最新解决结果意味着$\boldsymbol{\textsf{D}}_{\boldsymbol{\textsf{Che}}}(\boldsymbol{\textsf{T}}_\boldsymbol{\textsf{Λ}})$、$\boldsymbol{\textsf{λ}}_{\boldsymbol{\textsf{SG}}}(\boldsymbol{\textsf{T}}_\boldsymbol{\textsf{Λ}})$、$\boldsymbol{\textsf{λ}}_1(\boldsymbol{\textsf{Λ}}^*) \boldsymbol{\boldsymbol{\textsf{\textbf{≥}}}} \boldsymbol{\textsf{c}} \boldsymbol{\boldsymbol{\textsf{\textbf{>}}}} \boldsymbol{\textsf{0}}$，其中$\boldsymbol{\textsf{c}} \boldsymbol{\boldsymbol{\textsf{\textbf{>}}}} \boldsymbol{\textsf{0}}$是与维数$\boldsymbol{\textsf{n}}$无关的普适常数；这可视为对所有平坦环面的Kannan–Lovász–Simonovits猜想的正面解决。虽然存在格$\boldsymbol{\textsf{Λ}}$及对应的Voronoi胞腔$\boldsymbol{\textsf{K}} = \boldsymbol{\textsf{K}}_\boldsymbol{\textsf{Λ}}$，使得上述谱隙界不存在与维数无关的逆不等式，但我们证明在特定截面铺砌假设下，该不等式实际上是等价的（数值常数范围内）。
英文摘要
We record several elementary relations between spectral and isoperimetric parameters of a flat torus $\mathbb{T}_Λ= \mathbb{R}^n/Λ$ and the covariance structure of a fundamental domain $K$ for the lattice $Λ\subset \mathbb{R}^n$. For every measurable fundamental domain $K$ and nonzero vector $ξ$ in the dual lattice $Λ^*$, we observe the sharp directional variance estimate \[ \left\langle \operatorname{Cov}_K ξ,ξ\right\rangle \geq \frac{1}{12}. \] This yields a lower bound on the torus spectral gap $λ_{\mathrm{SG}}(\mathbb{T}_Λ)$ (equivalently, the length of the shortest nonzero dual vector $λ_1(Λ^*)$) in terms of the maximal covariance of $K$: \[ λ_{\mathrm{SG}}(\mathbb{T}_Λ) = 4π^2 λ_1(Λ^*)^2 \geq \frac{π^2}{3\left\|\operatorname{Cov}_K\right\|_{\mathrm{op}}}. \] Analogous sharp results are obtained for the isoperimetric profile and the Cheeger constant $D_{\mathrm{Che}}(\mathbb{T}_Λ)$ using an old argument of Hadwiger. In particular, when the Voronoi cell $K_Λ$ of a lattice with $\det Λ= 1$ is isotropic, the recent resolution of the Slicing Problem by Klartag and Lehec implies that \[ D_{\mathrm{Che}}(\mathbb{T}_Λ),\quad λ_{\mathrm{SG}}(\mathbb{T}_Λ),\quad λ_1(Λ^*) \geq c > 0, \] where $c > 0$ is a universal constant independent of dimension $n$; this may be thought of as a positive resolution of the Kannan--Lovász--Simonovits conjecture for all flat tori. While there are lattices $Λ$ and corresponding Voronoi cells $K = K_Λ$ for which no dimension-independent converse inequality to the spectral-gap bound above can hold, we show that under a certain sectional tiling hypothesis, this inequality is in fact an equivalence (up to numerical constants).
Comments11 pages, comments welcome
