$$\begin{aligned}
 & & S_n =\sum_{k=1}^\infty b_k e^{-k\alpha_n}\sin^2{(\pi uk)}
 \ge 4\sum_{k=1}^\infty b_k e^{-k\alpha_n}\parallel uk\parallel^2
 \nonumber \\
 & & \ge 4u^2\sum_{1\le k\le\pi/\alpha_n} k^2 b_k e^{-k\alpha_n}
 \ge (\delta_n/\pi)^2 \sum_{1\le k\le\pi/\alpha_n} k^2 b_k
 e^{-k\alpha_n} \nonumber \\
 & & =\frac{\alpha_n^{2+2\rho/3}}{(\pi\omega(n))^2}
 \sum_{1\le k\le\pi/\alpha_n} k^{\rho+1}\frac{k^2 b_k}{k^{\rho+1}}
 e^{-k\alpha_n} \nonumber \\
 & & =\frac{\alpha_n^{2+2\rho/3}}{(\pi\omega(n))^2} \sum_{1\le
 k\le\pi/\alpha_n} (k^{\rho+1} e^{-k\alpha_n})(b_k k^{-\rho+1}).
 \nonumber
 \end{aligned}$$ It is easy to check that the sequence $k^{\rho+1}e^{-k\alpha_n},
 k\ge 1$, is non-increasing if $k\ge(\rho+1)/\alpha_n-1/2+O(\alpha_n)$. Hence, for $\rho+1<\pi$, (\[wik\]) with $K=\pi,\rho+1$ implies that $$\begin{aligned}
 \label{snpi}
 & & S_n \ge\frac{\alpha_n^{2+2\rho/3}}{(\pi\omega(n))^2} \sum_{(\rho+1)/\alpha_n\le
 k\le\pi/\alpha_n} (k^{\rho+1} e^{-k\alpha_n})(b_k k^{-\rho+1})
 \nonumber \\
 & & \ge\frac{\alpha_n^{2+2\rho/3}}{(\pi\omega(n))^2}
 (\pi/\alpha_n)^{\rho+1} e^{-\pi}\sum_{(\rho+1)/\alpha_n\le
 k\le\pi/\alpha_n} b_k k^{-\rho+1} \nonumber \\
 & & =\frac{\pi^{\rho-1}e^{-\pi}\alpha_n^{1-\rho/3}}
 {\omega^2(n)}\left(\sum_{1\le k\le\pi/\alpha_n} b_k k^{-\rho+1}
 -\sum_{1\le k<(\rho+1)/\alpha_n} b_k k^{-\rho+1}\right) \nonumber \\
 & & \sim A\pi^{\rho-1}(\pi-\rho-1)e^{-\pi}\alpha_n^{-\rho/3}/
 \omega^2(n).
 \end{aligned}$$ If $\pi\le\rho+1$, then $k^{\rho+1}e^{-k\alpha_n}$ is a non-decreasing sequence for $1\le k\le\pi/\alpha_n$. Then, for some $l\in (0,\pi)$, $k^{\rho+1} e^{-k\alpha_n}\ge l^{\rho+1}
 e^{-l\alpha_n}$ and in the same way we observe that $$\begin{aligned}
\label{snro}
 & & S_n\ge \frac{\alpha_n^{2+2\rho/3}}{(\pi\omega(n))^2} \sum_{l/\alpha_n\le
 k\le\pi/\alpha_n} (k^{\rho+1} e^{-k\alpha_n})(b_k k^{-\rho+1})
 \nonumber \\
 & & \ge \frac{\alpha_n^{2+2\rho/3} l^{\rho+1} e^{-l}}{(\pi\omega(n))^2 \alpha_n^{\rho+1}}
 \sum_{l/\alpha_n\le k\le\pi/\alpha_n} b_k k^{-\rho+1}
  \nonumber \\
  & & \sim
 \frac{Al^{\rho+1}e^{-l}(\pi-l)\alpha_n^{-\rho/3}}
 {(\pi\omega(n))^2}(1+o(1)).
 \end{aligned}$$ Consequently, (\[realparts\]), (\[snpi\]) and (\[snro\]) imply that there are two constants $C_3,\epsilon_1>0$, such that $$\label{sn}
 S_n \ge C_3\alpha_n^{-\epsilon_1}$$ uniformly for $\delta_n/2\pi\le\mid u\mid<1/2$. Moreover, 