A dimension-uniform weak Gaussian bound for stable lattices

---

The theorem

A rank-$m$ Euclidean lattice $L$ is stable if $\det L=1$ and every nonzero sublattice has determinant at least one. Set

$$\theta_L(t)=\sum_{v\in L}e^{-\pi t^2\|v\|^2}.$$

Theorem. There are an absolute $T_0$ and a function $\epsilon(t)\downarrow0$, independent of rank, such that every stable lattice satisfies

$$\boxed{\theta_L(t)\leq e^{\epsilon(t)m}\quad(t\geq T_0).} \tag{1}$$

Consequently, for $0<u\leq1/T_0$,

$$\boxed{\theta_L(u)\leq u^{-m}e^{\epsilon(1/u)m}.} \tag{2}$$

This is an exponential-in-dimension bound, not a dimension-free bound such as $\theta_L(t)\leq3/2$ at fixed $t$. It is nevertheless sufficient for critical moment extraction.

1. The checked convex-body input

The regular $M$-ellipsoid theorem, at parameter $\alpha=1$, supplies an absolute $C\geq1$: every symmetric convex body $K\subset\mathbb R^m$ has a centered ellipsoid $E$ with

$$N(K,uE),\ N(E,uK)\leq e^{Cm/u}\qquad(u\geq C). \tag{3}$$

Here $N(F,H)$ is the least number of translates of $H$ covering $F$. The exact statement was checked in Vritsiou, Theorem 1, equations (2)--(3) (arXiv:2303.17753v2). It explicitly restates Pisier's 1989 theorem; this is not a new verification of Pisier's original proof.

For measurable $F\subset\mathbb R^m$, write $\gamma(F)=\int_F e^{-\pi\|x\|^2}\,dx$ and $\Gamma_t(F)=\gamma(tF)$. Define

$$c=\frac{e^{-1/2}}{2C},\qquad T_0=16eC^4,\qquad a(t)=\frac{t}{4e}-\frac{\log2}{2},$$ $$\epsilon(t)=\frac{C/c}{\sqrt t} -\log(1-e^{-a(t)})\qquad(t\geq T_0). \tag{4}$$

These constants are dimension-independent, $a(t)>0$, and $\epsilon(t)$ is positive and decreases to zero.

2. Every volume-one symmetric body has a good position

Claim. If $K=-K$ and $|K|=1$, then for each $t\geq T_0$ there is $A\in\mathrm{SL}_m(\mathbb R)$ with

$$\Gamma_t(AK)\geq e^{-\epsilon(t)m}. \tag{5}$$

Proof. Let $D_m=r_mB_2^m$ be the volume-one ball, and put $b=|E|^{1/m}$. Applying (3) at $u=2C$ and comparing volumes gives

$$1\leq e^{m/2}(2C)^m|E|,\qquad b\geq c.$$

Write $E=TB_2^m$ with $T$ positive definite. The map $A=br_mT^{-1}$ has determinant one and sends $E$ to $bD_m$. Put $R=\sqrt t$, $u=bt/R=b\sqrt t$. Since $t\geq T_0=(2C/c)^2$, one has $u\geq2C$. Invariance and scaling of covering numbers give

$$N(RD_m,tAK)=N(bD_m,(bt/R)AK) =N(E,uK) \leq e^{(C/c)m/\sqrt t}. \tag{6}$$

For every symmetric convex body $F$, its translates have Gaussian measure at most $\gamma(F)$. An elementary justification is as follows. For each centered ball $B$, put $U=(F+x)\cap B$. Symmetry and convexity give $(U-U)/2\subset F\cap B$; Brunn--Minkowski implies $|U|\leq|(U-U)/2|$. Integrate the resulting intersection-volume inequality over Gaussian level balls. A cover in (6) therefore yields

$$\gamma(RD_m)\leq N(RD_m,tAK)\,\Gamma_t(AK). \tag{7}$$

The radius $r_m>\sqrt{m/(2\pi e)}$. For example, $1=\int_{\mathbb R^m}e^{-\pi\|x\|^2}dx$ is strictly larger than the volume of a radius-$\sqrt{m/(2\pi)}$ ball times $e^{-m/2}$, which gives this bound after taking volume radii. If $G$ has density $e^{-\pi\|x\|^2}$, then $\mathbb E e^{\pi\|G\|^2/2}=2^{m/2}$. Markov's inequality gives

$$1-\gamma(\sqrt tD_m)\leq e^{-tm/(4e)}2^{m/2}=e^{-a(t)m}. \tag{8}$$

Equations (6)--(8), together with $1-z^m\geq(1-z)^m$ for $0<z<1$, prove (5). In particular, no equal-volume assumption on the regular ellipsoid has been inserted. $\square$

3. A fixed-parameter induction on stable Voronoi cells

Let $V(L)$ be the centered Voronoi cell of $L$. We prove

$$\Gamma_t(V(L))\geq e^{-\epsilon(t)m}. \tag{9}$$

The needed lattice facts are checked in Regev--Stephens-Davidowitz (arXiv:1611.05979v6): stable lattices form a compact set and a boundary lattice splits into proper stable sublattice and orthogonal quotient (Proposition 2.5); the Gaussian Voronoi functional has the position gradient of Theorem 3.1; Gaussian-isotropic bodies maximize Gaussian mass over determinant-one positions (Proposition 4.3). These apply at every positive parameter.

Fix $t\geq T_0$. Induct on rank, choosing a stable $L$ minimizing $\Gamma_t(V(L))$, which exists by compactness and continuity. In rank one, $L=\mathbb Z$ up to isometry and its volume-one interval has no other determinant-one position; (5) applies directly.

If the minimizer is also a local minimum among determinant-one lattices, apply the gradient identity to $f(x)=t^m e^{-\pi t^2x}$. Its derivative matrix is

$$-2\pi t^{m+2}\int_{V(L)} e^{-\pi t^2\|x\|^2}xx^{\mathsf T}\,dx .$$

Vanishing in all trace-zero directions makes this a scalar matrix. Thus $V(L)$ is Gaussian-isotropic at parameter $1/t$, and

$$\Gamma_t(V(L))\geq\Gamma_t(AV(L)) \quad\text{for every }A\in\mathrm{SL}_m(\mathbb R).$$

As $|V(L)|=\det L=1$, (5) proves (9) in this case. The lattice minimum has supplied stationarity, and body stationarity supplies a maximum over positions; there is no reversal of the desired inequality.

Otherwise $L$ lies on the stable boundary. There is a primitive sublattice $M$ of rank $0<h<m$ such that $M$ and $Q=L/M$, the orthogonal quotient, are stable. Their orthogonal product Voronoi body is a fundamental body for $L$. The nearest-point map from that body to $V(L)$ preserves volume and decreases norm almost everywhere, so

$$\Gamma_t(V(L))\geq\Gamma_t(V(M))\Gamma_t(V(Q)). \tag{10}$$

This is precisely the source's Claim 2.6 and Corollary 2.9, also displayed as equation (17). Induction at the same $t$ gives a lower bound $e^{-\epsilon(t)h}e^{-\epsilon(t)(m-h)} =e^{-\epsilon(t)m}$. This proves (9).

Finally $\theta_L(t)\Gamma_t(V(L))\leq1$, by the source's Lemma 4.1. It also follows directly by integrating translated Gaussian densities on the symmetric Voronoi cell and pairing $x$ with $-x$. Equation (9) proves (1).

4. Duality and review

The dual of a stable lattice is stable. One can check this by saturating a sublattice of $L^*$, taking its perpendicular sublattice in $L$, and using the complementary determinant identity at $\det L=1$. Poisson summation now gives

$$\theta_L(u)=u^{-m}\theta_{L^*}(1/u) \leq u^{-m}e^{\epsilon(1/u)m},$$

which is (2).

The primary passages, the volume normalization, covering rescaling, Gaussian translation and tail inequalities, and the full fixed-parameter induction are author-recorded; a separate reviewer checked each, and another checked the moment application. These are internal reviews, not external verification of an E0774 solution. No computation or constant-parameter Gaussian extremality conjecture enters the proof.