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It is well-known that for every $N \geq 1$ and $d \geq 1$ there exist point sets $x_1, \dots, x_N \in [0,1]^d$ whose discrepancy with respect to the Lebesgue measure is of order at most $(\log N)^{d-1} N^{-1}$.
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Algorithmic discrepancy beyond partial coloring
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C. Aistleitner and J. Dick · 2015
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An algorithm for Komlós conjecture matching Banaszczyk’s bound
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