Let $f: {\Bbb R}^n \longrightarrow {\Bbb R}$ be a positive definite quadratic form and let $y \in {\Bbb R}^n$ be a point. We present a fully polynomial randomized approximation scheme (FPRAS) for computing $\sum_{x \in {\Bbb Z}^n} e^{-f(x)}$, provided the eigenvalues of $f$ lie in the interval roughly between $s$ and $e^{s}$ and for computing $\sum_{x \in {\Bbb Z}^n} e^{-f(x-y)}$, provided the eigenvalues of $f$ lie in the interval roughly between $e^{-s}$ and $s^{-1}$ for some $s \geq 3$. To compute the first sum, we represent it as the integral of an explicit log-concave function on ${\Bbb R}^n$, and to compute the second sum, we use the reciprocity relation for theta functions. We then apply our results to test the existence of many short integer vectors in a given subspace $L \subset {\Bbb R}^n$, to estimate the distance from a given point to a lattice, and to sample a random lattice point from the discrete Gaussian distribution.

翻译：设$f: {\Bbb R}^n \longrightarrow {\Bbb R}$为正定二次型，$y \in {\Bbb R}^n$为一点。我们提出一个完全多项式随机近似方案（FPRAS），用于计算$\sum_{x \in {\Bbb Z}^n} e^{-f(x)}$，其中$f$的特征值大致位于区间$[s, e^{s}]$内；以及用于计算$\sum_{x \in {\Bbb Z}^n} e^{-f(x-y)}$，其中$f$的特征值大致位于区间$[e^{-s}, s^{-1}]$内，且$s \geq 3$。为计算第一个求和式，我们将其表示为${\Bbb R}^n$上一个显式对数凹函数的积分；为计算第二个求和式，我们利用theta函数的互反关系。随后，我们将结果应用于检验给定子空间$L \subset {\Bbb R}^n$中是否存在大量短整向量、估计给定点到某个格点的距离，以及从离散高斯分布中随机采样格点。