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We may assume the following theorem:

Theorem: A real number $\lambda$ is irrational iff the set $\{m+\lambda n\mid m,n\in\mathbb{Z}\}$ is a dense subset of $\mathbb{R}$.

Consider the points $$\gamma(t)=(a\cos t+b\sin t, a\sin t-b\cos t, c\cos(\lambda t)+d\sin(\lambda t), -c\sin(\lambda t)+d\cos(\lambda t))$$ for all $t\in\mathbb{R}$, for some constants $a,b,c,d$ where $a^2+b^2=c^2+d^2=1$.

Let $X=\{(x_1,x_2,x_3,x_4)\mid x_1^2+x_2^2=x_3^2+x_4^2=1\}$.

Why must the points $\gamma(t)$ be dense in $X$?

PJ Miller
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1 Answers1

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HINT: If $a=\cos\theta$ and $b=\sin\theta$, then the first two coordinates are $\cos(t-\theta)$ and $\sin(t-\theta)$
Then for any particular value of $t$, you add multiples of $2\pi$ and the third and fourth coordinates are dense in that circle.

Empy2
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