This question is motivated from this post of mine about the following problem:
Problem: Are there sets $A,B$ of integers such that $A\cap B=\emptyset$ and $A\times B\rightarrow\mathbb{Z}$, $(a,b)\mapsto a+b$ is bijective?
To try to solve this problem, I encountered with a positional numeral system called negabinary. The process that was naturally led to negabinary from a problem that seem not to be related at first glance is quite mysterious for me. So I want to continue exploring a little more. I especially want to know whether such a relationship is necessary.
I ask the following 3 questions:
Q1: Find (other) examples of $A,B$ satisfying the conditions in the problem above. Are they related with positional numeral systems?
Q2: How to define that $A,B$ are "related with" a positional numeral system? Is the definition below appropriate?
Q3: Is a relationship necessary? In other words, do there such $A,B$ which are not related with any positional numeral system exist?
For Q1, we already have an example in the previous post: $$ A=\{1+\sum_{i=0}^{N}a_{i}2^{2i}\;|\;a_{i}\in\{0,1\}\},\quad B=\{-1-\sum_{i=0}^{N}b_{i}2^{2i+1}\;|\;b_{i}\in\{0,1\}\}. $$ A little more generally, for any positive integer $r>1$, $$ A=\{1+\sum_{i=0}^{N}a_{i}r^{2i}\;|\;a_{i}\in\{0,1,\dotsc,r-1\}\},\quad B=\{-1-\sum_{i=0}^{N}b_{i}r^{2i+1}\;|\;b_{i}\in\{0,1,\dotsc,r-1\}\} $$ satisfies the conditions. Can we find examples of some other types?
For Q2, I'm currently using the following definition:
Definition. Let $A,B$ be sets of integers which satisfy the conditions in the problem above. We say $A,B$ are related with a positional numeral system in a base $q\in\mathbb{Z}$ ($|q|>1$), if there exist $\Lambda_{1},\Lambda_{2}\subset\mathbb{Z}_{\geq 0}$, $u_{1},u_{2}\in\mathbb{Z}$, and $\epsilon_{1},\epsilon_{2}\in\{-1,1\}$ such that $\mathbb{Z}_{\geq 0}=\Lambda_{1}\sqcup\Lambda_{2}$ (disjoint union), $u_{1}+u_{2}=0$, and $$ A=\{u_{1}+\epsilon_{1}\sum_{i\in\Lambda_{1}}a_{i}q^{i}\;|\;a_{i}\in\{0,1,\dotsc,|q|-1\}\},\quad B=\{u_{2}+\epsilon_{2}\sum_{i\in\Lambda_{2}}b_{i}q^{i}\;|\;b_{i}\in\{0,1,\dotsc,|q|-1\}\}. $$ For example, we have another example in the previous post: $$ A=\{a_{0}+\sum_{i=1}^{N}a_{i}2^{2i-1}\;|\;a_{0},a_{i}\in\{0,1\}\},\quad B=\{-1-\sum_{i=1}^{N}b_{i}2^{2i}\;|\;b_{i}\in\{0,1\}\}. $$ This is related with the negabinary by taking $q=-2$, $\Lambda_{1}=\{0,1,3,5,7,\cdots\}$, $\Lambda_{2}=\{2,4,6,8,\cdots\}$, $u_{1}=1$, $u_{2}=-1$, and $\epsilon_{1}=\epsilon_{2}=-1$ (here $a_{0}$ will be replaced with $1-a_{0}$). I'm thinking with this definition for the time being, but there may be other examples where we might want to change the definition in the future.
For Q3, I think a relationship is necessary, but for now I have only an emotional basis. I think the algorithm written in the previous post is something that one can naturally come up with when trying to solve the problem above, if he or she did not know any strategy based on a positional numeral system. Although we were able to prove precisely that the algorithm was related with the negabinary, when I look back, it still has a mysterious feeling. I feel there is something that necessarily leads to a positional numeral system in the problem itself. I want to know if this intuition is right or wrong.
Of course answers do not have to answer all questions. Answering partially is also welcome. Any ideas will be appreciated. Thank you!