Dyadic cubes of generation $k$ in ${\mathbb{R}}^n$

Fix $n\ge 1$. For $k\in \mathbb{N}$ and a function $m:n\to \mathbb{Z}$ (The integers as equivalence classes of pairs of naturals), whose values are read inside $\mathbb{R}$ along the canonical embedding, the dyadic cube of generation $k$ and index $m$ is the half-open box (Half-open boxes in ${\mathbb{R}}^n$ and their volume)

$Q_{k,m}:=B(a,b),a_i:=m_i{2}^{-k},b_i:=(m_i+1)2^{-k}(i<n),$

that is $Q_{k,m}=\{x\in {\mathbb{R}}^n:m_i{2}^{-k}<x_i\le (m_i+1)2^{-k}\text{ for every }i<n\}$, the powers being the integer powers of Integer powers $a^m$. A dyadic cube is a set of this form for some $k$ and $m$; its generation is $k$ and its side length is $2^{-k}$.

Every dyadic cube is nonempty, since $m_i{2}^{-k}<(m_i+1)2^{-k}$ for every $i$, so by Half-open boxes in ${\mathbb{R}}^n$ and their volume its parameters are determined by the set; the generation and the index are therefore determined by the cube as well. At $k=0$ the cubes are the translates of the unit cube $(0,1{]}^n$ by integer vectors, and $Q_{0,\mathbf{0}}=(0,1{]}^n$.

• The half-open convention is what makes the family a tiling. The cubes of a fixed generation are pairwise disjoint and cover ${\mathbb{R}}^n$ (For each generation, the dyadic cubes of that generation are pairwise disjoint and cover ${\mathbb{R}}^n$), which would fail for closed cubes, whose faces overlap, and for open cubes, which miss the grid points.

• Generations are natural numbers here, so the side lengths $2^{-k}$ are at most $1$ and there is a coarsest generation. Nothing below needs cubes larger than the unit cube, and bounding the generations below is what makes the maximal-cube selection of Every open subset of ${\mathbb{R}}^n$ is the union of a countable pairwise disjoint family of dyadic cubes work.