The Poisson moment assumption for Haar-distributed lattices

Let $\mathcal B$ be a symmetric set in $\bR^n$ of volume $V=2^{\mathcal O(n)}$, let $k=cn$ for a constant $0<c<1$, let $\mu$ be the Haar distribution on lattices, and let $Y$ be Poisson with mean $V/2$. Poisson moment assumption. The $k$-th moment satisfies

$$\mathbb E_{Y\sim\operatorname{Pois}(V/2)}[Y^k]\leq\mathbb E_{\Lambda\sim\mu}\left[\left(\frac{|\Lambda\cap\mathcal B|}{2}\right)^k\right]\leq\mathbb E_{Y\sim\operatorname{Pois}(V/2)}[Y^k]+o(1).$$

The Poisson moment is

$$\mathbb E_{Y\sim\operatorname{Pois}(V/2)}[Y^k]=e^{-V/2}\sum_{i=0}^{\infty}\frac{i^k}{i!}(V/2)^i.$$

This assumption is used to derive conditional list-decoding results for Haar lattices; it extends known averaging formulas to the moment range needed by the application.