In answer to the question as to whether $\text{tr}(C^n)$ is dense in $(-2,2)$:
Choose $z=\exp(2 \pi i \theta)$ where $\theta$ is irrational, and let $C$ be the diagonal matrix with $z$ and $\overline{z}$ on the diagonal. By Weyl's criterion , the fractional parts of $n \theta$ are equidistributed modulo 1, and thus $\{z^n\}$ is dense in the unit circle. From this it follows easily that $\text{Re}(z^n)$ is dense in $(-1,1)$.
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In answer to the question as to whether $\text{tr}(C^n)$ is dense in $(-2,2)$:
Choose $z=\exp(2 \pi i \theta)$ where $\theta$ is irrational, and let $C$ be the diagonal matrix with $z$ and $\overline{z}$ on the diagonal. By Weyl's criterion, the fractional parts of $n \theta$ are equidistributed modulo 1, and thus |
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