We want to consider the Gaussian distribution on a full-rank lattice \(\Lambda \subseteq \mathbb{R}^{n}\) via the function \(\rho_{s}:\mathbb{R}^{n} \rightarrow \mathbb{R}\) given by \[\begin{equation} \rho_{s}(x) = \exp( - \pi |x|^{2}/s^{2}), \end{equation}\] for some parameter \(s >0\).
We define \[\begin{equation} \rho_{s}(\Lambda) = \sum_{v \in \Lambda} \rho_{s}(v) \end{equation}\] and consider the distribution \[\begin{equation} D_{\Lambda,s} = \sum_{v \in \Lambda} D_{\Lambda,s}(v) \delta_{v}, \end{equation}\] where \[\begin{equation} D_{\Lambda,s}(v) = \rho_{s}(v)/\rho_{s}(\Lambda). \end{equation}\]
One wants to understand what happens when we project the distribution modulo \(q \in \mathbb{Z}_{\geq 1}\). One has the map \[\begin{equation} \pi: \Lambda \rightarrow \Lambda/q \Lambda \simeq (\mathbb{Z}/q\mathbb{Z})^{n}. \end{equation}\] This gives a push forward measure \(D_{\mathbb{Z}^{n}/q \mathbb{Z}^{n}}\) on \(\mathbb{Z}^{n}/q\mathbb{Z}^{n}\).
One has for the uniform distribution \(\mathcal{U}_{q}\) on \(\mathbb{Z}^{n}/q\mathbb{Z}^{n}\) the estimate \[\begin{equation} \| D_{\mathbb{Z}^{n}/q\mathbb{Z}^{n}} - \mathcal{U}_{q}\|_{\mathrm{TV}} \leq \tfrac{1}{2}\left( \rho_{\frac{q}{s}}(\Lambda^{\mathrm{dual}}) - 1\right) \end{equation}\]
Hence, it makes sense to estimate the right hand side even for random lattices.
See here for some related ideas. The original reference is [1].
This page was updated on May 21, 2026.
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