On the general dyadic grids in $\mathbb{R}^d$

TL;DR

This paper extends the characterization of adjacent dyadic systems from the real line to higher dimensions, revealing that $d+1$ such systems are needed in $ eal^d$, with geometric structures underpinning their adjacency.

Contribution

It provides a complete geometric description of adjacent dyadic systems in all dimensions, generalizing previous one-dimensional results and identifying the minimal number of systems needed.

Findings

$d+1$ dyadic systems are necessary and sufficient for adjacency in $ eal^d$

Adjacency can be characterized by projections onto coordinate axes

The geometric structures underlying these systems are rich and warrant further study

Abstract

Adjacent dyadic systems are pivotal in analysis and related fields to study continuous objects via collections of dyadic ones. In our prior work (joint with Jiang, Olson and Wei) we describe precise necessary and sufficient conditions for two dyadic systems on the real line to be adjacent. Here we extend this work to all dimensions, which turns out to have many surprising difficulties due to the fact that $d+1$, not $2^d$, grids is the optimal number in an adjacent dyadic system in $\mathbb{R}^d$. As a byproduct, we show that a collection of $d+1$ dyadic systems in $\mathbb{R}^d$ is adjacent if and only if the projection of any two of them onto any coordinate axis are adjacent on $\mathbb{R}$. The underlying geometric structures that arise in this higher dimensional generalization are interesting objects themselves, ripe for future study; these lead us to a compact, geometric description of our main result. We describe these structures, along with what adjacent dyadic (and $n$-adic, for any $n$) systems look like, from a variety of contexts, relating them to previous work, as well as illustrating a specific example.