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Lattices in the sense of discrete subgroups of Euclidean spaces, as used in number theory, discrete geometry, Lie groups, etc. (Not to be confused with lattice theory or lattices as used in physics! For lattices (ordered sets), use the tag: [lattice-theory])
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Higher dimensional analogs of logarithmic density
The natural $k$-dimensional analogue of logarithmic density is
$$
\lim_{x \rightarrow \infty} \frac{1}{(\log x)^{k}} \sum_{\substack{n_1, \ldots, n_k \leq x \\ (n_1, \ldots, n_k) \in S}} \prod_{i = 1} …