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$\mathbb{Z}\subset A=\mathbb{Z}\subset \mathbb R^2$. In this case $\partial\mathbb{Z}= \mathbb{Z}$ and $\mathop{Leb}^+(\mathbb Z)=\infty$ and $H^1(\mathbb Z)=0$.

You can get a bounded example of the same type. Take a countable nowhere dense set $A$ in the unit disc such that the $\varepsilon$-neighborhood of $A$ contains a disc of radius $\sqrt[3]{\varepsilon}$.

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$\mathbb Q\subset \mathbb{Z}\subset \mathbb R^2$.

$\mathop{Leb}^+(\mathbb Q)=\infty$ Z)=\infty$and$H^1(\mathbb Q)=0$.Z)=0$

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