6 problems

Let $d\geq2$, and let $\mathcal{E}$ be a $\frac12$-sparse antichain of $d$-dimensional dyadic rectangles. Let $\operatorname{sh}(\mathcal{E})$ be the shadow of $\mathcal{E}$. Antic…

Let $\Phi:\mathbb{Z}^{d-1}\to\mathbb{Z}$ be monotone increasing in each argument, and let $\mathcal{Z}_d$ be the collection of dyadic rectangles in $\mathbb{R}^d$ whose sidelengths…

Let $Q\subset\mathbb{R}^d$ be a cube, let $\mathrm{BMO}_{\varepsilon}^{\mathrm{dyad}}(Q)$ denote the dyadic BMO class with parameter $\varepsilon$, and define … An extension…

Let $N\in\mathbb{N}$ with $N\geq 2$ and let $0\leq \ell\leq N-2$. For the dyadic collection ${\mathcal E}^N_{\ell,0}$ and the postorder rearrangement operator $\tau_N$, write…

Let $M=\{m_j:j\geq1\}$ be a shift, and set $N_j(M)$ for the associated Semenov counting quantity. A sequence is called decomposable when it satisfies the hypothesis of Definition 1…

Let $\t$ be a rearrangement of the dyadic intervals, and write $\cF_1,\dots,\cF_N$ for subcollections of the dyadic intervals. Suppose there is a constant $C$ such that, for every…