Discrepancy-order conjecture for golden-ratio Hammersley point sets

Let

denote the golden ratio, let $F$ be the Fibonacci sequence used to define the point-set size, and let $H_m$ be the Hammersley point set in base

containing $N=F^m$ points, where $m e 0$ is an integer. The quantities $L_{2,N}(H_m)$ and $D_N^{*}(H_m)$ denote, respectively, the $L_2$ discrepancy and star discrepancy.

Discrepancy-order conjecture. For $m e 0$,

$$L_{2,N}(H_m)=\Theta\left(\frac{\log N}{N}\right)$$

and

$$D_N^{*}(H_m)=\Theta\left(\frac{\log N}{N}\right).$$

The claim is motivated by numerical evidence from the discrepancy plots and is stated as strong evidence for the displayed convergence; no proof or resolution is supplied in the paper.