For every subset $A$ of real numbers, we call $B\subseteq \mathbb{R}$ a sub-factor related to $A$ if:
(1) $(A-A)\cap (B-B)=\{ 0\}$,
(2) $(A-A)+B=\mathbb{R}$.
We can show that every subset $A$ has related sub-factors. Now, my questions are:
(a) Is there a subset $A$ with a related finite and an infinite sub-factors $B_1$, $B_2$, respectively?
(b) Is it true that for every two cardinal numbers $2\leq\alpha\leq \beta\leq 2^{\aleph_0}$ there exists a subset $A$ with two sub-factors $B_1, B_2$ satisfying $|B_1|=\alpha$ and $|B_2|=\beta$
(it is clear that (a) is satisfied if (b) is true).
Note that $B-B=\{ b-\beta:b,\beta\in B\}$, $A+B=\{a+b:a\in A,b\in B\}$, and see
Two uncountable subsets of real numbers without any interval and two relations ,
https://www.worldscientific.com/doi/abs/10.1142/S0219498820501017?journalCode=jaa.
Thanks in advance.