Given an irrational number $\alpha > 0$ (I specifically care about the case where $\alpha$ is a quadratic integer which is also a Pisot number, but the question can be asked in general), it is well known that if we define $c_m = \{ m\alpha\}$ where $\{ x\} = x - \lfloor x \rfloor$ is the fractional part of $x$, then the sequence $( c_m)_{m\geq 1}$ is equidistributed in $[0,1)$.
Now, define two sequences $(a_n)_{n\geq 1}$, $(b_n)_{n\geq 1}$ as follows:$$ a_n = \max_{1 \leq m \leq n} c_m, $$$$ b_n = \min_{1\leq m \leq n} c_m. $$
These are a non-decreasing and a non-increasing sequence, respectively. The question is: can you recover the number $\alpha$ if you only know at which indices the sequences increase/decrease, but do not know the values of the sequences?
That is, do the sets $A_\alpha = \{ k \in \mathbb{N} : a_{k} > a_{k-1} \}, \, B_\alpha = \{ k \in \mathbb{N} : b_k < b_{k-1} \}$ uniquely specify $\{ \alpha \}$ in the general case (they do not uniquely specify $\alpha$ since you can always take the trivial counterexamples $\alpha' = \alpha + M$, for $M\in \mathbb{N}$)? And in the particular case where $\alpha$ is a quadratic (irrational) Pisot number, do $A_\alpha$ and $B_\alpha$ uniquely specify $\alpha$ (among all such numbers)?
Any comments or pointers to relevant results would be appreciated, thanks!
