Topological and Diophantine properties of lattice subset projections

Fix $1\le n<m,k=m-n\mathrm{.}$ The Grassmannian $Gr(n,m)$ is a compact $kn$-dimensional manifold with a unique rotation invariant probability measure ${\sigma }_n\mathrm{.}$ For $W\in Gr(n,m)$, $P_W:{\mathbb{R}}^m\mapsto W$ is orthogonal projection. A lattice subset $L\subset {\mathbb{Z}}^m\subset {\mathbb{R}}^m$ is called $k$-dense if it intersects $C(O):={\bigcup }_{V\in O}V\mathrm{\backslash }\{0\}$ for every nonempty open $O\subset Gr(k,m)$. We use Baire’s category theorem [4] to prove that $L$ is $k$-dense iff $L_{n,lim}:=\{W\in Gr(n,m):0isalimitpointof{P}_W(L)\}$ is a $G_{\delta }$ set. We use Khintchine-Groshev’s theorem [5,13,20] to characterize Diophantine properties of $L_{n,lim}$ by lacunary properties of $L$ and construct $k$-dense $L$ with ${\sigma }_n(L_{n,lim})=0$ and with ${\sigma }_n(L_{n,lim})=1.$ We pose related questions about the construction of multidimensional crystalline measures and Fourier quasicrystals.