---

Source. Lemma 3.30, p. 19, of Cabrelli, Lacey, Molter and Pipher, Variations on the theme of Journé's lemma, in the edition named on the source card.

Statement

Setting (p. 18). With $\operatorname{Enl}_2(\mathcal U)$ as in (3.20), $\{M\mathbf 1_{\operatorname{sh}(\mathcal U)}>\frac1{16}\}$, and $\operatorname{Dil}_{(\mu_1,\mu_2)}R=\mu_1R_{(1)}\times\mu_2R_{(2)}$ (1.3), the embeddedness is

$$\operatorname{emb}(R,\mathcal U)=\sup\{\mu_1\mu_2:\operatorname{Dil}_{(\mu_1,\mu_2)}R\subset\operatorname{Enl}_2(\mathcal U),\ \mu_1,\mu_2\ge1\}.$$

The paper notes (p. 19) that this can be essentially smaller than the equal-dilation embeddedness of Lemma 3.23.

Lemma 3.30 (p. 19, quoted). "In the case $d=2$, for any $\epsilon>0$, any collection of rectangles $\mathcal U$ in the plane, whose shadow has finite measure, and all $\mathcal U'\subset\mathcal U$ of rectangles which are maximal, we have

$$\sum_{R\in\mathcal U'}\operatorname{emb}(R,\mathcal U')^{-\epsilon}\lvert R\rvert\lesssim\lvert\operatorname{sh}(\mathcal U')\rvert.$$

The implied constant depends only on $\epsilon>0$."

The printed display weights by $\operatorname{emb}(R,\mathcal U')$, not $\operatorname{emb}(R,\mathcal U)$, and the statement does not say within which collection the rectangles of $\mathcal U'$ are maximal. A footnote on p. 18 says this formulation had not yet found application in the literature.

Proof pointer

P. 19, a sketch only. The standard reduction is refined by fixing $(\mu_1,\mu_2)$ with $\mu\le\mu_1\mu_2\le2\mu$, each $\mu_j\ge1$, and assuming every $R$ satisfies $\operatorname{Dil}_{(\mu_1/2,\mu_2/2)}R\subset\operatorname{Enl}_2(\mathcal U)$ but not $\operatorname{Dil}_{2(\mu_1,\mu_2)}R$; there are $\lesssim(\log\mu)^3$ such classes, and the paper says the argument of Section 3.2 then proceeds with only modest changes.

Dependencies

None in the corpus. Read depth: claims checked; the definition and statement were read clause by clause on pp. 18--19. The paper gives no full proof. Nothing here is independently reviewed.

Bears on

The paper names no Erdős problem.