<s>[INST] Bounding a sum involving a $\Re((z\zeta)^N)$ term This is a follow up to this question. Any help would be very much appreciated.
Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ or some other $N>ak^2$.
Let $\zeta:=\exp(2\pi i/k)$ and $\alpha_v:=\zeta^v+\zeta^{-v}+\zeta^{-1}$.
Here there are five questions of varying intricacy. An answer to four is what I am hoping to achieve myself but an answer to an earlier part should go a long way towards helping and obviously an answer to part 5. would be amazing.
I have given a fairly trivial bound below which is good for my needs. If I don't get a better answer by the end of the bounty period I will accept my own (CW) answer and grant charMD the bounty.

  
*
  
*Simplify, where $v\in\{1,2,\dots,(k-1)/2\}$, $$1-\sin^2\left(\frac{2\pi
 v}{k}\right)\Re\left((\alpha_v\zeta)^N\right)+|\alpha_v|^{2N}.$$
  
*Upper bound, where $v\in\{1,2,\dots,(k-1)/2\}$, $$\sec^2\left(\frac{2\pi v}{k}\right)\left(1-\sin^2\left(\frac{2\pi
 v}{k}\right)\Re\left((\alpha_v\zeta)^N\right)+|\alpha_v|^{2N}\right)\leq
 f_2(v,k,N).$$
  
*Simplify, where $v\in\{1,2,\dots,(k-1)/2\}$, $$\sec^2\left(\frac{2\pi v}{k}\right)\left(1-\sin^2\left(\frac{2\pi
 v}{k}\right)\Re\left((\alpha_v\zeta)^N\right)+|\alpha_v|^{2N}\right).$$
  
*Upper bound $$\frac{1}{4^{2N-1}}\sum_{v=1}^{\frac{k-1}{2}}\sec^2\left(\frac{2\pi v}{k}\right)\left(1-\sin^2\left(\frac{2\pi
 v}{k}\right)\Re\left((\alpha_v\zeta)^N\right)+|\alpha_v|^{2N}\right)\leq f_4(k,N).$$
  
*Sum $$\frac{1}{4^{2N-1}}\sum_{v=1}^{\frac{k-1}{2}}\sec^2 
\left(\frac{2\pi\,v}{k}\right)\left(1-\sin^2\left(\frac{2\pi
 v}{k}\right)\Re\left((\alpha_v\zeta)^N\right)+|\alpha_v|^{2N}\right).$$

 [/INST] Consider first 
$$\begin{align}
1-\sin^2\left(\frac{2\pi
 v}{k}\right)\Re\left((\alpha_v\zeta)^N\right)+|\alpha_v|^{2N}&\leq 1+\sin^2\left(\frac{2\pi
 v}{k}\right)\left|\Re\left((\alpha_v\zeta)^N\right)\right|+|\alpha_v|^{2N}
\\&\leq 1+\sin^2\left(\frac{2\pi
 v}{k}\right)\left|(\alpha_v\zeta)^N\right|+|\alpha_v|^{2N}
\\&\leq 1+\sin^2\left(\frac{2\pi
 v}{k}\right)\left|\alpha_v\right|^N+|\alpha_v|^{2N}
\\&\leq 1+\sin^2\left(\frac{2\pi
 v}{k}\right)3^N+3^{2N}
\end{align}$$
In terms of efficiency, I am interested in $k$ large but $N=\mathcal{O}(k^2)$ and for $N\approx \frac{k}{2}\mod k$,
$$-\Re\left((\alpha_v\zeta)^N\right)\approx +\Re\left((\alpha_v)^N\right).$$
The largest problem is that 
$$\alpha_v=2\cos\left(\frac{2\pi v}{k}\right)+\zeta^{-1}$$
has a large real part for $k$ large and $v$ small but as $v\rightarrow \frac{k-1}{2}$
$$\alpha_v\approx -1,$$
rather than $\alpha_v\approx 3$ as is the case for $v$ small. Hopefully this doesn't make the bound unusable (should find out soon).
Therefore
$$
\begin{align} &\frac{1}{4^{2N-1}}\sum_{v=1}^{\frac{k-1}{2}}\sec^2\left(\frac{2\pi v}{k}\right)\left(1-\sin^2\left(\frac{2\pi
 v}{k}\right)\Re\left((\alpha_v\zeta)^N\right)+|\alpha_v|^{2N}\right)\\ &\leq \frac{1}{4^{2N-1}}\sum_{v=1}^{\frac{k-1}{2}}\sec^2\left(\frac{2\pi v}{k}\right)\left(1+\sin^2\left(\frac{2\pi
 v}{k}\right)3^N+3^{2N}\right)
\\&=\frac{1}{4^{2N-1}}\left((1+3^{2N})\sum_{v=1}^{\frac{k-1}{2}}\sec^2\left(\frac{2\pi v}{k}\right)+3^{N}\sum_{v=1}^{\frac{k-1}{2}}\tan^2\left(\frac{2\pi v}{k}\right)\right).
\end{align}$$
Now using $\sec^2A=1+\tan^2A$ and this answer of Joriki this is equal to
\begin{align*}
\frac{1}{4^{2N-1}}\left((1+3^{2N})\left(\frac{k-1}{2}+\frac{k(k-1)}{2}+3^{N}\frac{k(k-1)}{2}\right)\right)&=2\frac{k-1}{4^{2N}}\left[(k+1)3^{2N}+k\cdot 3^N+k+1\right]
\end{align*}
Any sharpening would be most welcome in an answer. An ideal answer would be a good bound of the form:
$$\frac{1}{4^{2N-1}}\sum_{v=1}^{\frac{k-1}{2}}\sec^2\left(\frac{2\pi v}{k}\right)\left(1-\sin^2\left(\frac{2\pi
 v}{k}\right)\Re\left((\alpha_v\zeta)^N\right)+|\alpha_v|^{2N}\right)\leq f(N,k)e^{-\pi^2(2N-1)/k^2},$$
with the 'smaller' $f(N,k)$ the better. This answer here leads to an
$$f(N,k)=6(k^2-1)\left(\frac{3}{4}e^{\pi^2/k^2}\right)^{2N}$$
</s>