In this book on page 82 I found an estimate $$\sum\_{j=1}^{n-2}\frac{\sin(k+1)\theta\_{j}}{\sin\theta\_{j}}=O\_{\epsilon}\left( p^{k\epsilon}\right)$$ as $k$ goes to infinity, for all $\epsilon>0$. Here $p$ is a prime number and $\theta\_{1},...,\theta\_{n-2}$ are complex numbers. After this estimate, the authors says: The last clearly implies that $\theta\_{j}$ $\left(1\leq\theta\_{j}\leq n-2\right)$ are real numbers. Why is this clearly true?


P.S. I hope this edited question is now ok.



On an asymptotic in Sarnak's book: "Some applications of modular forms"
A much more detailed writeup of this is is Davidoff/Sarnak/Vallette, see page 127 and on, especially p. 130.


