Subsets of real numbers satisfying the two conditions

Question

Is there any uncountable subset $B$ of real numbers such that:

(1) $(B-B)\cap (-1,1)=\{ 0\}$,

(2) $(-1,1)+B=\mathbb{R}$?

Also, what is the answer if $(-1,1)$ is replaced by $[-1,1)$, $(-1,1]$ or $[-1,1]$?

Note that $B=\mathbb{Z}$ satisfies (1) and (2) but it is countable, and $B-B=\{ b-\beta:b,\beta\in B\}$, $A+B=\{a+b:a\in A,b\in B\}$.

Answer 1

This fails immediately because of (1): there is no uncountable subset $B$ of $\mathbb R$ such that $(B-B) \cap (-1,1)=\{0\} $ because every uncountable subset has an accumulation point. Adding endpoints to the interval doesn't change anything: there are pairs of points with arbitrarily small positive differences in $B$.