Computing the theta function

Abstract: Let $f: {\Bbb R}ⁿ \longrightarrow {\Bbb R}$ be a positive definite quadratic form and let $y \in {\Bbb R}<sup>n$ be a point. We present a fully polynomial randomized approximation scheme (FPRAS) for computing $\sum_{x \in {\Bbb Z}^n} e<sup>{-f(x)}$, provided the eigenvalues of $f$ lie in the interval roughly between $s$ and $e<sup>{s}$ and for computing $\sum_{x \in {\Bbb Z}^n} e<sup>{-f(x-y)}$, provided the eigenvalues of $f$ lie in the interval roughly between $e<sup>{-s}$ and $s<sup>{-1}$ for some $s \geq 3$. To compute the first sum, we represent it as the integral of an explicit log-concave function on ${\Bbb R}<sup>n$, 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 $L \subset {\Bbb R}<sup>n$, to estimate the distance from a given point to a lattice, and to sample a random lattice point from the discrete Gaussian distribution.