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A lattice subset $L \\subset \\mathbb Z^m \\subset \\mathbb R^m$ is called $k$-dense if it intersects $C(O) := \\bigcup_{V \\in O} V\\backslash \\{0\\}$ for every nonempty open $O \\subset Gr(k,m)$. We use Baire's category theorem [4] to prove that $L$ is $k$-dense iff $L_{n,lim} := \\{W \\in Gr(n,m) : 0 \\mbox{ is a limit point of } P_W(L) \\}$ is a $G_\\delta$ set. 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A lattice subset $L \\subset \\mathbb Z^m \\subset \\mathbb R^m$ is called $k$-dense if it intersects $C(O) := \\bigcup_{V \\in O} V\\backslash \\{0\\}$ for every nonempty open $O \\subset Gr(k,m)$. We use Baire's category theorem [4] to prove that $L$ is $k$-dense iff $L_{n,lim} := \\{W \\in Gr(n,m) : 0 \\mbox{ is a limit point of } P_W(L) \\}$ is a $G_\\delta$ set. 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