Minimum L2-star discrepancy in the unit hypercube
Place n points in [0,1]^d. For every axis-parallel box anchored at the origin, compare the box's volume with the fraction of points it contains; integrate the square of that error over all boxes. That integral is the squared L2-star discrepancy. Make it as small as possible.
Formal definition
- ContainerThe d-dimensional unit hypercube [0,1]^d, coordinates in the closed interval
- SubmissionExactly n points, each d decimal coordinates; coincidences are allowed
- ObjectiveMinimize the L2-star discrepancy given by Warnock's closed formula; the verifier computes its square exactly on the nine-decimal grid
- ScoringThe record is the exact integer M·D² with the common denominator cleared; the page shows D, rounded up at the twelfth decimal
Getting a feel for it
What it measures
With a finite budget for rendering or numerical integration, sample points have to imitate the uniform distribution. Star discrepancy asks which origin-anchored box is systematically over- or under-sampled. The L∞ version (P54) punishes the single worst box; this one punishes the mean square over all boxes — which is what lets it climb into higher dimensions while the verifier gets simpler.
Where the frontier is
Provably optimal L2-star point sets for fixed (n, d) barely exist: the literature settles only n = 1 and 2 exactly. Every sub-problem here is open.
SourceCurrent best solutions by n
Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.
Discussion (0) ↓Discussion
Talk strategy, share methods, ask why you are stuck. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. The floor is earned: break a record once, anywhere, and it is yours for good. New posts appear after an automated review.
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