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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\Omega \\subseteq \\R^n$ be an open set whose complement $F = \\R^n \\setminus \\Omega$ is nonempty.\nFor a set $E$ write $\\operatorname{diam}(E)$ for its diameter and $\\operatorname{dist}(E, F)$ for its distance to $F$.\nThen there is a countable family $(Q_k)$ of closed cubes with sides parallel to the axes such that:\n\n1. $\\Omega = \\bigcup_k Q_k$;\n2. the interiors of the cubes $Q_k$ are pairwise disjoint;\n3. for every $k$,\n$$\\operatorname{diam}(Q_k) \\le \\operatorname{dist}(Q_k, F) \\le 4 \\operatorname{diam}(Q_k).$$\n\n## Remarks\n\nThe cubes can be chosen dyadic, that is, of the form $2^{-j}([0, 1]^n + m)$ with $j \\in \\Z$ and $m \\in \\Z^n$.\nProof idea: for each $j$, take the dyadic cubes of side $2^{-j}$ that meet the layer $\\set{x \\in \\Omega : 2^{-j+1}\\sqrt{n} \u003c \\operatorname{dist}(x, F) \\le 2^{-j+2}\\sqrt{n}}$.\nThese cubes satisfy clause 3 by the triangle inequality, and they cover $\\Omega$.\nTwo dyadic cubes are either nested or have disjoint interiors, so keeping only the maximal cubes in the collection gives clause 2.\n\nClause 3 implies that neighbouring cubes have comparable sizes: if $Q_k$ and $Q_l$ touch, then $\\operatorname{diam}(Q_l) \\le \\operatorname{dist}(Q_l, F) \\le \\operatorname{dist}(Q_k, F) + \\operatorname{diam}(Q_k) \\le 5 \\operatorname{diam}(Q_k)$.\nIt follows that, for a fixed small enlargement factor such as $1 + \\frac{1}{4}$, each point of $\\Omega$ lies in at most $C(n)$ of the enlarged cubes.\n\nExample: for $\\Omega = (0, \\infty) \\subseteq \\R$ and $F = (-\\infty, 0]$, the intervals $[2^{j}, 2^{j+1}]$ for $j \\in \\Z$ cover $\\Omega$ with disjoint interiors.\nTheir distance to $F$ equals their length, so clause 3 holds.\n\n## Explanations\n\n### L1 human\n\nAn open set can be tiled by cubes whose size is proportional to their distance from the boundary.\nNear the boundary the cubes are tiny, far from it they are large, like a mosaic that gets finer and finer as it approaches the edge.\nThe proportionality makes the tiling well adapted to the shape of the set.\n\n### L2 human\n\nWhitney introduced this decomposition to extend smooth functions from a closed set $F$ to all of $\\R^n$, in the Whitney extension theorem.\nIn harmonic analysis it is applied to the open set $\\set{Mf \u003e \\lambda}$ in the Calderón–Zygmund decomposition, in the atomic decomposition of Hardy spaces, and in proofs of good-$\\lambda$ inequalities.\nA partition of unity subordinate to the slightly enlarged cubes has derivatives bounded by $C \\operatorname{dist}(x, F)^{-k}$, which is what such arguments need.\n",
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