The signature sequence of a positive irrational number $\theta$ is defined as follows:
- Sort the set $\mathbb{Z_+ \times Z_+}$ via the function $f(c,d)=c+d\theta$.
- Create a sequence from the first number of each ordered pair of this sequence
For example, A084531 is the signature sequence of $\phi$.
My question is, can there be two unequal numbers $\theta_1$, $\theta_2$ with the same signature sequence?
No, you can recover $\theta$ from its signature sequence. First, note that you can recover the relation $a+b\theta<c+d\theta$ on $\mathbb{Z}_+\times\mathbb{Z}_+$ from the signature sequence, since $a+b\theta<c+d\theta$ iff the number $a$ appears $b$ times in the signature sequence before the number $c$ appears $d$ times. In particular, the set of pairs $(a,b)\in\mathbb{Z}_+\times\mathbb{Z}_+$ such that $a+1+\theta<1+(b+1)\theta$ can be recovered from the signature sequence. But this is exactly the set of pairs $(a,b)$ such that $a/b<\theta$. Since $\theta$ is the supremum of the set of all positive rational numbers less than $\theta$, this determines $\theta$.