The discrepancy lower-bound conjecture in dimensions at least three

Let $\mathcal A_N$ be an arbitrary set of $N$ points in $[0,1]^d$, and define its discrepancy function by

$$D_N(x)=\#\bigl(\mathcal A_N\cap[0,x)\bigr)-N|[0,x)|,$$

where $[0,x)=\prod_{j=1}^d[0,x_j)$. Discrepancy lower-bound conjecture. For $d\ge3$, every such point set satisfies

$$\|D_N\|_{L^\infty}\gtrsim(\log N)^{d/2}.$$

This would improve Roth's universal $(\log N)^{(d-1)/2}$ lower bound by a factor of $\sqrt{\log N}$. The assertion is known in dimension two, has partial results in dimension three, and the paper proves only an exponent $(d-1)/2+\eta$ for some $\eta(d)>0$ in dimensions at least three.