Discrete tori and trigonometric sums
type: Item
instance of: scholarly article
title: Discrete tori and trigonometric sums
zbMATH Open document ID: 1502.60117
author: A. A. Grigor'yan
author: Yong Lin
author: Shing-Tung Yau
published in: The Journal of Geometric Analysis
publication date: 2022-10-04T00:00:00Z
review text: For \(v=(v_1,\cdots, v_n)\in \mathbb{Z}^n\) set \(|v|=|v_1|+\cdots +|v_n|\) and \(\bar{v}=(|v_1|,\cdots, |v_n|)\). For \(v\in \mathbb{Z}^n_+\) set \(v!=v_1!\cdots v_n!\). Fix an integer valued \(n\times n\) matrix \(M\) with \(m:=\text{det}M>1\). Let \(M^*\) be the transpose of \(M\). For any non-negative integer \(s\) set \[ C_s(M)=\sum_{v\in M\mathbb{Z}^n, z\in \mathbb{Z}^n_+, |v|+2|z|=s} \frac{s!}{z! (\bar{v}+z)!}. \] The main result of this paper states that for the torus \[ W=(M^*)^{-1}\mathbb{Z}^n/\mathbb{Z}^n \] and for any non-negative integer \(s\), it holds that \[ \sum_{w\in W}\left(\sum_{k=1}^n \cos 2\pi w_k \right)^s=\frac{m}{2^s}C_s(M). \]
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 11L03
Mathematics Subject Classification ID: 05C81
Mathematics Subject Classification ID: 60J10
zbMATH DE Number: 7595997
zbMATH Keywords: trace formula
zbMATH Keywords: heat kernel
zbMATH Keywords: Markov chain
zbMATH Keywords: discrete torus
zbMATH Keywords: trigonometric sum
MaRDI profile type: Publication
full work available at URL: s12220-022-01051-6
OpenAlex ID: W4296844542
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DOI: 10.1007/S12220-022-01051-6
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description: scientific article; zbMATH DE number 7595997