Construction of a small but fat set?

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Is it possible to find a subset $A$ of the real line $\mathbb R$ such that the Lebesgue measure of $A$ minus its interior is positive ?

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What about $(\Bbb R-\Bbb Q)\cap [0,1]$

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A fat Cantor set. A is closed, its interior is empty, so $A$ minus its iterior is $A$ itself. See previous question: https://math.stackexchange.com/a/287872/442