Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
124 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
5 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Computing the theta function (2208.05405v2)

Published 10 Aug 2022 in math.NA, cs.CG, cs.DS, cs.NA, and math.CO

Abstract: 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.

Summary

We haven't generated a summary for this paper yet.