In this paper, there is the following claim (Pg. 1850):

If $1 - \eta(w) \ne 0$, then $|\omega| \ge \rho$. In that case, $\omega$ belongs to $D^{\varepsilon^3}\theta^\ast$ for $\lesssim \rho^{-n} D^{n-2 + n\varepsilon^3}$ different $\theta$. (For comparison, note that the total number of $\theta$ is $D^{n-1}$.) Note that the $\rho^{-n}$ might not be optimal, but since we take $\rho^{-1}$ to be very small compared to $D$, it does not matter as long as it is a polynomial power.

Question: How would you show the bound $\lesssim \rho^{-n} D^{n-2 + n\varepsilon^3}$?

For context, the unit sphere $S^{n-1}$ is covered by $1/D$-caps $\theta$, where $D$ is a large number. $\theta^*$ is defined to be a slab through the origin perpendicularly the direction of the cap $\theta$ (for each cap $\theta$, we call the outer normal direction of the center of $\theta$ on $S^{n-1}$ as the direction of $\theta$.) Also, $\rho := D^{\varepsilon^3} S^{-1}$ is defined on Pg. 1849, where $S = D^{\varepsilon/10n}$ for some tiny $\varepsilon > 0$.

I do not think more context from the paper would be required to establish the claim above. Perhaps a picture or some sort of semi-visual explanation would help see what's going on. Thank you!

In this paper, there is the following claim (Pg. 1850):

If $1 - \eta(w) \ne 0$, then $|\omega| \ge \rho$. In that case, $\omega$ belongs to $D^{\varepsilon^3}\theta^\ast$ for $\lesssim \rho^{-n} D^{n-2 + n\varepsilon^3}$ different $\theta$. (For comparison, note that the total number of $\theta$ is $D^{n-1}$.) Note that the $\rho^{-n}$ might not be optimal, but since we take $\rho^{-1}$ to be very small compared to $D$, it does not matter as long as it is a polynomial power.

Question: How would you show the bound $\lesssim \rho^{-n} D^{n-2 + n\varepsilon^3}$?

For context, the unit sphere $S^{n-1}$ is covered by $1/D$-caps $\theta$, where $D$ is a large number. $\theta^*$ is defined to be a slab through the origin perpendicularly the direction of the cap $\theta$ (for each cap $\theta$, we call the outer normal direction of the center of $\theta$ on $S^{n-1}$ as the direction of $\theta$.) Also, $\rho := D^{\varepsilon^3} S^{-1}$ is defined on Pg. 1849, where $S = D^{\varepsilon/10n}$ for some tiny $\varepsilon > 0$.

I do not think more context from the paper is required to establish the claim above. Perhaps a picture or some sort of semi-visual explanation would help see what's going on. Thank you!