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Fibonacci sets and symmetrization in discrepancy theory

机译：斐波那契数列和差异理论中的对称化

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We study the Fibonacci sets from the point of view of their quality with respect to discrepancy and numerical integration. Let (b_n)_(n=0)~∞ be the sequence of Fibonacci numbers. The b_n-point Fibonacci set F_n c [0,1]~2 is defined as F-n := {(μ/6_n, {μb_(n-1)/b_n})}_(μ=1)~(bn) where {x} is the fractional part of a number x ∈ R. It is known that cubature formulas based on the Fibonacci set F_n give optimal rate of error of numerical integration for certain classes of functions with mixed smoothness. We give a Fourier analytic proof of the fact that the symmetrized Fibonacci set F_n' = F_n ∪ ((p1, 1 - p_2) : (p_1,p_2) ∈ F_n} has asymptotically minimal L_2 discrepancy. This approach also yields an exact formula for this quantity, which allows us to evaluate the constant in the discrepancy estimates. Numerical computations indicate that these sets have the smallest currently known L_2 discrepancy among two-dimensional point sets. We also introduce quartered L_p discrepancy, which is a modification of the L_p discrepancy symmetrized with respect to the center of the unit square. We prove that the Fibonacci set F_n has minimal in the sense of order quartered L_p discrepancy for all p ∈ (1, ∞). This in turn implies that certain two-fold symmetrizations of the Fibonacci set F_n are optimal with respect to the standard L_p discrepancy.
机译：我们从差异和数值积分的质量角度研究斐波那契数列。令（b_n）_（n = 0）〜∞为斐波那契数的序列。 b_n点斐波那契集合F_n c [0,1]〜2定义为Fn：= {（μ/ 6_n，{μb_（n-1）/ b_n}）} _（μ= 1）〜（bn）其中{x}是数x∈R的小数部分。众所周知，基于Fibonacci集F_n的算子公式对于某些具有混合平滑度的函数给出了最佳的数值积分误差率。我们给出一个傅立叶分析证明，即对称的斐波那契集F_n'= F_n∪（（p1，1-p_2）：（p_1，p_2）∈F_n}具有渐近最小的L_2差异。数值计算表明，这些点集在二维点集中具有最小的当前已知L_2差异；我们还引入了四分之一的L_p差异，这是对L_p差异的修正关于单位p的中心对称，我们证明Fibonacci集F_n在所有p∈（1，∞）的四分之一L_p阶差的意义上具有极小值，这反过来又意味着斐波那契集合F_n的某些对称性斐波那契集合F_n相对于标准L_p差异而言是最佳的。

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