Average decay of Fourier transforms and integer points in polyhedra
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 \[ \widehat{\chi_\mathcal P}(\xi) = \int_{\mathcal P} e^{-2\pi i\xi\cdot x} dx \qquad (\xi\in\mathbb R^k). \]
scientific article; zbMATH DE number 1185540