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Is there a measurable set $U\subset \Bbb R$ of Lebesgue measure 0 satisfying the property: For any two points $p, q\in \Bbb R$, there exists $a\in \Bbb R$ such that $\{p+a, q+a\}\subset U$.

Is there a measurable set $U\subset \Bbb R^2$ of measure 0 satisfying the property: For any two points $P, Q\in \Bbb R^2$, there exists $a\in \Bbb R$ such that $\{P, Q\} +(a, a)\subset U$.

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