Star discrepancy lower-bound conjecture in arbitrary dimension

Prove that for every dimension d and every finite point sequence x_0, ..., x_{N-1} in the unit cube, there exists a positive constant c_d depending only on d such that the star discrepancy satisfies D^*(x_0, ..., x_{N-1}) \ge c_d(\log N)^{d-1}/N.

Background

The paper states the standard conjectured lower bound for the star discrepancy of finite point sets in dimension d. This bound would establish the optimal order of discrepancy up to dimension-dependent constants and logarithmic factors, thereby explaining why low-discrepancy sequences achieve essentially the best possible asymptotic convergence order for quasi-Monte Carlo integration.

The conjecture is noted to have been proved for dimensions d\le 2 but remains unproved in general. It motivates the paper’s discussion of low-discrepancy sequences and the error bounds used for Sobol-based quasi-Monte Carlo methods.