8 problems

Let $d\ge 3$, let $\mathcal D^d$ be the dyadic rectangles in $[0,1]^d$, and let $h_R$ denote the tensor-product Haar function associated with a rectangle $R$. For an integer $n$, w…

Let $\mathcal{D}$ be the dyadic rectangles in $[0,1)^d$, and for each $R\in\mathcal{D}$ let $h_R$ denote its associated Haar function. For signs $\varepsilon_R\in\{-1,1\}$, conside…

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…

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

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…

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…