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Results tagged with lattices
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user 17408
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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Deformations of null-vectors of an integral unimodular lattice
Since all odd integral unimodular lattices are equivalent up to an $SO(n,n)$, then the lattice $\Lambda_{\rm sL} \oplus \Lambda_{\rm sL}$ is related to the 46-dimensional hypercubic lattice. …
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Is this related to a simple property of a lattice?
I am looking for a certain notion of sparseness of lattices.
I want to find a vector in $\mathbb{Z}^N$ that the minimal possible inner product with all the vectors of a given lattice. … I would expect that unimodular lattices with no roots (vectors of norm 1 or 2) are sparse in the above sense; the intuition being that they miss many of the short vectors in $\mathbb{Z}^N$. …
3
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Is this bounded from below?
Let $u_1, u_2, u_3 \in \mathbb{Z}$ such that $u_1^2 + u_2^2 = u_3^2$.
Is $(u_3 + \frac{u_1 + u_2}{\sqrt{2}})^2$ bounded from below?
The irrationality of $\sqrt{2}$ certainly precludes zero, but can …