论文标题
晶格点靠近海森堡球
Lattice Points Close to the Heisenberg Spheres
论文作者
论文摘要
我们研究了由海森伯格组引起的球体的晶格计数问题。特别是，我们证明了由各向异性规范产生的单位球上和附近的点数上的上限，$ \ |（z，z，t）\ |_α=（| z |^α+ | t |^|^{α/2}）作为第一步，我们将计数问题减少到界限能量积分之一。出现的主要新挑战是存在消失的曲率和不均匀的扩张。在此过程中，我们建立了由这些规范产生的表面测量的傅立叶变换的界限。此外，我们利用此处开发的技术来估计两个这样的表面交集中的晶格点数。
We study a lattice point counting problem for spheres arising from the Heisenberg groups. In particular, we prove an upper bound on the number of points on and near large dilates of the unit spheres generated by the anisotropic norms $\|(z,t)\|_α= ( |z|^α+ |t|^{α/2})^{1/α}$ for $α\geq 2$. As a first step, we reduce our counting problem to one of bounding an energy integral. The primary new challenges that arise are the presence of vanishing curvature and uneven dilations. In the process, we establish bounds on the Fourier transform of the surface measures arising from these norms. Further, we utilize the techniques developed here to estimate the number of lattice points in the intersection of two such surfaces.
