= Geodesic-cap and Carleson-box comparison
For a short arc $I$ of normalized length $\ell$, every point of its <hyperbolic geodesic cap> lies within Euclidean distance $C\ell$ of its center boundary point. Conversely, for any fixed $b>0$, the region with angular projection in $I$ and depth $1-|z|\le b\ell$ lies in $Q(\kappa I)$ for a fixed enlargement $\kappa$ depending only on $b$, when $I$ is sufficiently short. To verify this, rotate the midpoint to one and write the endpoints as $e^{\pm i\alpha}$. The cap inequality is $2\operatorname{Re}z>(1+|z|^2)\cos\alpha$. Its radial depth at the midpoint is $1-\sec\alpha+\tan\alpha$, comparable to $\alpha$ for small $\alpha$; its angular projection lies in $I$. Enlarging $\alpha$ by a sufficiently large fixed factor makes the inequality hold throughout the indicated rectangle. Large arcs use a finite cover of the disc. Thus <Carleson boxes> and <hyperbolic geodesic caps> define equivalent <Carleson measure> conditions.
The rectangle containment can also be checked directly. Write $z=re^{i\theta}$ with $|\theta|\le\pi\ell$ and $1-r\le b\ell$. Its membership in the enlarged cap is equivalent to
$$
\cos\theta-\cos(\kappa\pi\ell)>\frac{(1-r)^2}{2r}\cos(\kappa\pi\ell).
$$
For small enough $\ell$, $r\ge1/2$. The right side is at most $b^2\ell^2$, while the left side is at least $(\kappa^2-1)\pi^2\ell^2/8$, using the sine product formula and $\sin x\ge x/2$ for small positive $x$. Thus $\kappa=4+4b/\pi$ suffices.
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