Minimum star discrepancy in the unit square

Formal definition

• ContainerThe unit square [0, 1]², with lower-left corner at (0, 0) and upper-right corner at (1, 1)
• SubmissionExactly n distinct points, each coordinate written as a decimal string with at most nine decimal places
• RectanglesTest rectangles are half-open boxes [0, u) × [0, v) with (u, v) ∈ [0, 1]²; points on the top or right boundary count as outside
• Star discrepancyThe indicator 𝟙 is 1 when the point lies in [0, u) × [0, v) and 0 otherwise; D*(P) is the supremum absolute difference between the point fraction and the area uv$$D^{\ast }\left(P\right)=\underset{(u,v)\in {\left[0,1\right]}^2}{\sup }\left|\frac{1}{n}\sum _{i=1}^n𝟙\left[x_i<u,y_i<v\right]-uv\right|$$
• ObjectiveMinimize D*(P) over all valid point sets P. The verifier evaluates the open and closed one-sided limits on the grid induced by the submitted coordinates and compares exact integers$$\underset{P}{\min }{D}^{\ast }\left(P\right)$$