Average decay of Fourier transforms and integer points in polyhedra
type: Item
instance of: scholarly article
title: Average decay of Fourier transforms and integer points in polyhedra
zbMATH Open document ID: 0937.11043
DOI: 10.1007/BF02559969
author: Luca Brandolini
author: Leonardo Colzani
author: Giancarlo Travaglini
published in: Arkiv för Matematik
publication date: 1998-11-08T00:00:00Z
review text: The main topic of this article is the ``average'' number of (integer) lattice points in a polyhedron \(\mathcal P\) in \({\mathbb R}^k\) (with characteristic function \(\chi_\mathcal P\)) which is supposed to be submitted to an arbitrary rotation \(\theta\in SO(k)\), a shift by a vector \(\mathbf t\in\mathbb R^k/\mathbb Z^k\), and a dilation (``blowing up'') by a large parameter \(\rho\in\mathbb R\). In a natural way the discrepancy (``lattice rest'') is defined as \[ D(\rho,\theta,\mathbf t) = \sum_{\mathbf n\in\mathbb Z^k} \chi_{\rho \theta^{-1}\mathcal P - \mathbf t} (\mathbf n) \;-\;\rho^k \text{ vol}(\mathcal P) \] (number of lattice points minus volume). Although this discrepancy may be as large as \(\rho^{k-1}\), the authors' main target is to provide results which illustrate that, ``with high probability'', it is only of logarithmic order. In particular, it is shown that \[ \begin{aligned}\sup_{\lambda>0}\left(\lambda\mid \{\theta\in SO(k),\;\mathbf t\in\mathbb R^k/\mathbb Z^k:\;|D(\rho,\theta,\mathbf t)|>\lambda\}|\right) &\ll (\log\rho)^{k-1} , \\ \int_{SO(k)}\int_{\mathbb R^k/\mathbb Z^k} |D(\rho,\theta,\mathbf t)|d\mathbf t d\theta &\ll (\log\rho)^k, \\ \left(\int_{SO(k)}\int_{\mathbb R^k/\mathbb Z^k} |D(\rho,\theta,\mathbf t)|^p d\mathbf t d\theta \right)^{1/p} &\ll \rho^{(k-1)(1-1/p)}\qquad(1<p\leq\infty)\end{aligned} \] This is achieved by a skillful analysis of the behaviour of the Fourier transform \[ \widehat{\chi_\mathcal P}(\xi) = \int_{\mathcal P} e^{-2\pi i\xi\cdot x} dx \qquad (\xi\in\mathbb R^k). \]
reviewed by: Werner Georg Nowak
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 1185540
zbMATH Keywords: lattice points
zbMATH Keywords: polyhedra
zbMATH Keywords: Fourier transforms
zbMATH Keywords: lattice rest
zbMATH Keywords: discrepancy
zbMATH Keywords: average number
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02559969
OpenAlex ID: W2009174488
description: scientific article; zbMATH DE number 1185540