Laboratory

白昼之光，岂知夜色之深。

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Again, into deeper.

there you dear.

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Here, for a lattice $L \subset \mathbb{R}^{n}$ and $s>0$,

(1)

\begin{align} \rho_{s}(L):=\sum_{\boldsymbol{y} \in L} e^{-\pi\|\boldsymbol{y}\|^{2} / s^{2}} \end{align}

is the Gaussian mass of the lattice with parameter $s$. This can be seen as a smooth version of the point-counting function $r \mapsto\left|L \cap r B_{2}^{n}\right|$, with the parameter $s$ playing the role of the radius $r$, and it arises naturally in a number of contexts (often in the form of the theta function, $\Theta_{\mathcal{L}}(i y):=\rho_{1 / \sqrt{y}}(\mathcal{L})$ ). In particular, Theorem 1.2 immediately implies that $\left|L \cap r B_{2}^{n}\right| \leq 3 e^{\pi t^{2} r^{2}} / 2$ for any radius $r>0$. (We note that the constant $3 / 2$ in this bound and the theorem statement is chosen for convenience, and a similar statement holds with any constant strictly bigger than 1.)