10 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 $d\ge 2$. Let $\mathcal D$ be the dyadic intervals in $[0,1)$, let $\mathcal D^d$ be the dyadic rectangles in $[0,1)^d$, and let $h_R$ be the associated multiparameter Haar fun…

Signed small ball inequality. There is a constant $C'_d$, depending only on the dimension, such that

Small ball inequality. There is a constant $C_d$ such that

Let $d\ge 3$, let $\mathcal D^d$ denote the dyadic rectangles in $[0,1]^d$, and for each $R\in\mathcal D^d$ let $h_R$ be the product of the $L^\infty$-normalized Haar functions on…

Let $d\ge 1$, let $\mathcal D^d$ be the dyadic rectangles in $[0,1]^d$, and let $h_R$ be the $L^\infty$-normalized Haar function associated with a rectangle $R$. For an integer…

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…

Small ball inequality conjecture. The displayed inequality should hold with the exponent $(d-2)/2$ in place of the trivial exponent $(d-1)/2$. This conjecture predicts the optimal…