4 problems

Let $d\ge 2$. Let $\mathcal D^d$ be the dyadic rectangles in $[0,1)^d$, and let $h_R$ be the associated multiparameter Haar function. For each $n\in\mathbb N_0$, consider rectangle…

Let $\mathcal A_N$ be an arbitrary set of $N$ points in $[0,1]^d$, and define its discrepancy function by … where $[0,x)=\prod_{j=1}^d[0,x_j)$. Discrepancy lower-bound conjecture.…

Smooth small ball inequality conjecture. The displayed inequality should hold, with an implied constant depending only on $d$ and $\varphi$. This is a smooth analogue of the Haar s…

Small ball inequality conjecture. For every $d\ge3$, the displayed inequality should hold. It is a central conjecture in the paper, with consequences for irregularities of distribu…