Let $X_0,X_1\in [0,1]$ and $b_1,b_2>0$ be integers. We are going to create a numeration system for vectors $(X_0,X_1)$, the base being the vector $(b_1,b_2)$, as follows.

Recursively define $X_k=\{b_2 X_{k-1} + b_1 X_{k-2}\}$, for $k>1$. Here $\{\cdot\}$ represents the fractional part function and $X_k\in [0,1]$. Clearly, $$d_k=b_1 X_k + b_2 X_{k+1} - X_{k+2}$$ is an integer between $0$ and $b_1+b_2-1$. The sequence $d_0, d_1, d_2,d_3,\cdots$ represents, by definition, the digits of $(X_0,X_1)$ in base $(b_1,b_2)$. If $b_1=0$ then the digits are just the standard digits of $X_1$ in base $b_2$.

**Questions:**

- Can two different vectors $(X_0,X_1)$ and $(X_0',X_1')$ have the exact same digits in base $(b_1,b_2)$, assuming $b_1,b_2>0$?
- Can you reconstruct $(X_0,X_1)$ if you only know its digits in base $(b_1,b_2)$?

My guess is that the answer to the first question is yes. So all that suffices is to provide an example. This would lead to a negative answer to my second question.

However, if the answer to the first question is negative, there would be the following interesting consequences. Let $b=b_1+b_2$. To each $(X_0,X_1)$ corresponds a unique number $f(X_0,X_1)\in[0,b]$ defined by its expansion in base $b$ as follows:

$$f(X_0,X_1)=\sum_{k=0}^\infty \frac{d_k}{b^{k}}.$$

The two consequences would be:

- Since for the immense majority of couples $(X_0,X_1)$ the distribution of the digits $d_k$ is NOT uniform on the set $\{0,1,2,\cdots,b-1\}$ (see below why), the number $f(X_0,X_1)$ is not normal. Since the set of non-normal numbers has zero Lebesgue measure, we mapped $[0,1]^2$ onto a set of Lebesgue measure zero. The mapping is bijective.
- We created an order on $[0,1]^2$. It is defined as follows: $(X_0,X_1) < (X_0',X_1')$ if and only if $f(X_0,X_1) < f(X_0',X_1')$.

**Some useful results**

In order to prove or disprove my claims, I offer the following result. While at this stage I strongly believe that the formula below is correct, I did not technically prove it. This is just based on pattern recognition techniques and experimental math, yet I think the proof should be easy.

$$X_k = \{A(k) X_1\} \mbox{ with } A(0) =\frac{X_0}{X_1}, A(1) =1, \mbox{ and } A(k)= b_2 A(k-1) + b_1 A(k-2).$$

More about this in my former MO question, here. In addition, as previously discussed, the digits of $(X_0, X_1)$ are almost surely NOT uniformly distributed over $\{0,1,\cdots b-1\}$, unlike classic digits of (say) $\log 2$ in base $b$. Just to give you an example (again based on strong empirical evidence but not a proof) this is the standard distribution of the digits in base $(b_1=3, b_2=3)$:

- digit $0$ appears with frequency $1/18$
- digit $1$ appears with frequency $3/18$
- digit $2$ appears with frequency $5/18$
- digit $3$ appears with frequency $5/18$
- digit $4$ appears with frequency $3/18$
- digit $5$ appears with frequency $1/18$

Essentially these are the frequencies you would observe in that base if you picked up $X_0,X_1$ randomly.