For all $n\in\mathbb{N}$, define $$\Delta(n) = \left\{F\left(k\ln(2), \frac{2\pi l}{2^k}\right),\, k\in\{0,\ldots,n\},\, l\in\{1,\ldots,2^k\}\right\}.$$ Fix $r>0$ as in question 6.6. Show that there exists a constant $A\geq 1$ satisfying the following two properties:
for all $g\in\Gamma(n\ln(2))$, $$|\{v\in\Delta(n) \text{ such that } d(gv_0,v)\leq r\}| \leq A,$$
for all $v\in\Delta(n)$, $$|\{g\in\Gamma(n\ln(2)) \text{ such that } d(gv_0,v)\leq r\}| \leq A.$$
For all $n\in\mathbb{N}$, define
$$\Delta(n) = \left\{F\left(k\ln(2), \frac{2\pi l}{2^k}\right),\, k\in\{0,\ldots,n\},\, l\in\{1,\ldots,2^k\}\right\}.$$
Fix $r>0$ as in question 6.6. Show that there exists a constant $A\geq 1$ satisfying the following two properties:
\begin{enumerate}
\item for all $g\in\Gamma(n\ln(2))$,
$$|\{v\in\Delta(n) \text{ such that } d(gv_0,v)\leq r\}| \leq A,$$
\item for all $v\in\Delta(n)$,
$$|\{g\in\Gamma(n\ln(2)) \text{ such that } d(gv_0,v)\leq r\}| \leq A.$$
\end{enumerate}