这是一份umass麻省大学 MATH 300作业代写的成功案例
Proof. Recall the notation for continued fractions. Since $\alpha$ is a quadratic irrational, there exists a minimal integer $p$, called the period of $\alpha$, such that $\alpha=\left[a_{1}, a_{2}, \ldots, a_{p}, a_{1}, a_{2}, \ldots, a_{p}, a_{1}\right.$, $\left.a_{2}, \ldots, a_{p}, \ldots\right]$. That is, $a_{k+i}=a_{i}$ if $k$ is a multiple of $p$.
Hence,
$$
q_{k+i}=a_{i} q_{k-1+i}+q_{k-2+i}, \quad 0 \leq i \leq p
$$
We estimate the difference of the last two series. Observe that
$$
\sum_{i=q_{k}+1}^{q_{n+k}} \chi_{I_{U}}\langle i \alpha\rangle=\sum_{i=q_{j}+1}^{q_{n+j}} \chi_{I_{V}}\langle i \alpha\rangle .
$$
MATH 300 COURSE NOTES ：
The entropy of $\Gamma$ verifies the inequality
$$
S(\Gamma) \leq S_{t}(\Gamma):=\log \frac{2|\Gamma|}{|\partial K|}+\frac{\beta}{e^{\beta}-1}
$$
where
$$
\beta=\log \frac{2|\Gamma|}{2|\Gamma|-|\partial K|}>0
$$
Proof: Maximize the function
$$
S\left(p_{1}, p_{2}, \ldots\right)=\sum_{n=1}^{\infty} p_{n} \log \frac{1}{p_{n}}
$$