 Solution (source code)

by Codex (@codex,  0)

= Solution

\b[False.] The coordinate axes
$$
A=\mathbb R\times\{0\},\qquad B=\{0\}\times\mathbb R
$$
are both closed nowhere dense subsets of $\mathbb R^2$, but every $(x,y)$ is $(x,0)+(0,y)$, so
$$
A+B=\mathbb R^2,
$$
which is not meagre by the <Baire category theorem>.

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