论文标题
兰伯特投影下的扰动晶格的球形帽差异
Spherical cap discrepancy of perturbed lattices under the Lambert projection
论文作者
论文摘要
鉴于任何完整的等级晶格和自然数n，我们认为由兰伯特图下的单位正方形与单位球体相交的尺度晶格所述的点，并表明其球体盖差的大多数是n的大多数，具有明确的领先系数，并且仅取决于晶格。使用引理建立证明，该引理与某些曲线的相交数量与瓷砖r^2的基本域的相交数量界定，甚至允许晶格的局部扰动而不会影响界限，证明对数值应用是稳定的。一个特殊情况使最新的确定性算法的帽差差异的前项产生了最小的常数。
Given any full rank lattice and a natural number N , we regard the point set given by the scaled lattice intersected with the unit square under the Lambert map to the unit sphere, and show that its spherical cap discrepancy is at most of order N , with leading coefficient given explicitly and depending on the lattice only. The proof is established using a lemma that bounds the amount of intersections of certain curves with fundamental domains that tile R^2 , and even allows for local perturbations of the lattice without affecting the bound, proving to be stable for numerical applications. A special case yields the smallest constant for the leading term of the cap discrepancy for deterministic algorithms up to date.
