Timeline for Is there a matrix C so that the trace of C^n is dense in R?
Current License: CC BY-SA 2.5
10 events
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| Feb 12, 2010 at 1:12 | history | edited | Autumn Kent | CC BY-SA 2.5 |
rescaling is bogus.; added 46 characters in body
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| Feb 12, 2010 at 0:56 | comment | added | Victor Miller | @GL: You have a point, I'll have to think about it. | |
| Feb 12, 2010 at 0:32 | comment | added | George Lowther | Yes, cos(nt) is dense in [-1,1] when theta is irrational. However, on its own that doesn't imply that 2 r^n cos(nt) either is or isn't dense in the reals. If theta is algebraic then you can use Roth's theorem to deduce that cos(nt) can't be less than (1/n^2) infinitely often, so 2r^ncos(nt)-> infinity. However, there is still the transcendental theta case. | |
| Feb 12, 2010 at 0:23 | comment | added | Victor Miller | I think yes. Write $t=2 \pi \theta$. There are two cases: 1) $\theta$ is rational -- then $\cos(nt)$ takes on only a finite set of values, and unless $\theta =$ an odd integer$/4$, they are non-zero. 2) $\theta$ is irrational. Then by Weyl's criterion, $\cos(nt)$ is dense in $(-1,1)$ | |
| Feb 12, 2010 at 0:17 | comment | added | George Lowther | Victor, if it has complex eigenvalues and r>1 then the traces are 2r^n cos(nt). Does this have to grow with n? | |
| Feb 12, 2010 at 0:12 | comment | added | Victor Miller | Here's a way around it: Let $r$ be the absolute value of the largest eigenvalue. If $r > 1$ then the traces are a sequence growing like $r^$ (perhaps times a polynomial in $n$). It's not hard to see that this sequence can't be dense in $\mathbb{R}^n$. If $r < 1$ it's also clear that the sequence can't be. Similarly for $r=1$. | |
| Feb 12, 2010 at 0:03 | comment | added | David E Speyer | How do you reduce to the determinant 1 case? | |
| Feb 11, 2010 at 23:23 | comment | added | Tom LaGatta | Is there an elliptic C such that tr(C^n) is dense in (-2,2)? | |
| Feb 11, 2010 at 22:34 | history | edited | Autumn Kent | CC BY-SA 2.5 |
rescaled matrix.
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| Feb 11, 2010 at 22:26 | history | answered | Autumn Kent | CC BY-SA 2.5 |