For the previous question: Is there a measurable set $U⊂\Bbb R$ of Lebesgue measure $0$ satisfying the property: For any two points $p,q∈\Bbb R$ , there exists $a∈\Bbb R$ such that $\{p+a,q+a\}\subset U$.
A very nice answer was given by Geyer(thank for the answer!)
More general question: Is there a measurable set $U⊂\Bbb R$ of Lebesgue measure $0$ satisfying the property that for any three points $p,q, r∈\Bbb R$ , there exists $a∈\Bbb R$ satisfying $\{p+a,q+a, r+a\}⊂U$. More generally, for any finite set of points $p_1, \ldots, p_n\Bbb R$, there exists $a\in \Bbb R$ such that $\{p_1+a,p_2+a,\ldots, p_n+a\}⊂U$.