The greedy star-discrepancy conjecture for the sequence generated by Algorithm 1

Let $\mathcal{S}^*$ be the sequence generated by Algorithm 1 in $[0,1)$, and let $D_N^*(\mathcal{S}^*)$ denote the star discrepancy of its first $N$ elements. Let $\mathcal{V}$ be the van der Corput sequence in base $2$.

Greedy star-discrepancy conjecture. The sequence $\mathcal{S}^*$ should be a low-discrepancy sequence, meaning that

$$D_N^*(\mathcal{S}^*)=\mathcal{O}(\log N).$$

Moreover,

$$\limsup_{N\to\infty}\frac{D_N^*(\mathcal{S}^*)}{\log N}<\limsup_{N\to\infty}\frac{D_N^*(\mathcal{V})}{\log N}=\frac{1}{3\log 2}=0.480898\ldots.$$

The conjecture is motivated by numerical comparisons showing that the greedy sequence has smaller and less fluctuating star discrepancy than the van der Corput sequence for the tested range. Proving the asserted optimal-order bound and the strict comparison remains open.