Optimal quadrature points on the torus · d = 3, n = 8

Formal definition

• ContainerThe d-dimensional torus: coordinates modulo 1, written in [0, 1)
• SubmissionExactly n points, each d decimal coordinates; coincidences are allowed
• Bernoulli polynomialThis problem fixes the second Bernoulli polynomial B₂$$B_2\left(t\right)=t^2-t+\frac{1}{6}$$
• KernelBraces denote the fractional part (coordinate differences modulo 1). The kernel fixes the coefficient at 6; it belongs to the same family as periodic L2 discrepancy (coefficient 3) and diaphony (coefficient 2π²), with a different weight$$K(x,y)=\prod _{r=1}^d\left(1+6B_2\left(\left\{x_r-y_r\right\}\right)\right)$$
• ScoreE(P) is the squared worst-case error of the equal-weight quadrature rule; the kernel integrates to 1 over the torus, so E(P) ≥ 0$$E\left(P\right)=\frac{1}{n^2}\sum _{i,j}K(x_i,x_j)-1$$
• ScoringThe record is stored as the exact integer n²S²ᵈ·E after clearing denominators; the page displays the error √E, rounded up at the twelfth decimal
• ObjectiveMake E(P), equivalently √E(P), as small as possible over all legal point sets P$$\underset{P}{\min }E\left(P\right)$$