# Questions tagged [lattices]

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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### Obstruction in construction of some lattices, related with $K3$ surfaces

I am considering a certain $K3$ surface that is lattice-polarized in two ways. This leads to the following simple problem in lattice theory: (Let me borrow notations for lattice from Ch.14 of this ...
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### Minimal Norm Vectors in certain Cyclotomic Ideal Lattices

Let $q$ be an odd prime and let $\zeta$ be a primitive $q^{th}$ root of unity. Let $I$ be the ideal in $\mathbb{Z}[\zeta]$ generated by $1-\zeta$. It is known that we can give $I$ the structure of an ...
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### Finite index subgroup of $\mathbb{Z}^2$ that is invariant under a non-singular matrix

Let $M$ be a matrix in $\operatorname{GL}(2, \mathbb{Z})$ that has at least one eigenvalue of absolute value strictly bigger than $1$. What are the finite index subgroups $H$ of $\mathbb{Z}^2$ such ...
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### Optimal covering trails in 3 and 4 dimensions

A couple of years ago, I constructively solved (inside the $AABB$ $[0,3]$ X $[0,3]$ X ... X $[0,3]$) the $k$-dimensional generalization of the infamous Nine-Dot Problem by S. Loyd (see Cyclopedia of ...
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### The smallest volume possible for a lattice with integer distances?

Let $\Lambda \subset\mathbb{R}^n$ be a lattice satisfying $\|x-y\|_2^2 \in \mathbb{Z}$ for all $x,y\in\Lambda$. How small can $\text{vol}(\Lambda)=\det(\Lambda)$ be? For example, in dimension $2$, the ...
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### Stability of successive minima with respect to the metric on the space of lattices

Let $\mathcal L$ be the space of unimodular (covolume one) lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the ...
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### Geometric explanation for coincidence in lengths of 16-dimensional even unimodular lattices?

Question Up to equivalence, there are two positive-definite even unimodular lattices in $16$ dimensions: $D_{8}^{+}\oplus D_{8}^{+}$ and $D_{16}^{+}$. As observed by Witt in 1941, the theory of ...
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### The range of each of successive minima for all unimodular lattices

Let $\mathcal L$ be the space of unimodular (covolume one) lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the ...
1 vote
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### Affine semigroup generating a lattice

This is a cross-post from MSE. Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$ of a Euclidean space). Assume that ...
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### Is there an exact solution for the number of points within a circle of radius r for an honeycomb lattice?

I want to ask if exists an exact solution for the number of points within a circle of radius r for an honeycomb lattice. I know that it is exist for an square lattice https://mathworld.wolfram.com/...
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### Orders of Niemeier lattice stabilizers

What is the order of the stabilizer of each Niemeier lattice, scaled so that its shortest nonzero vectors form a subset of those of the Leech lattice, within the latter’s automorphism group?
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### Is $(\omega+1)^\omega/{\cal U}$ "unique"?

If ${\cal U}_i$ free ultrafilters on $\omega$ for $i = 1,2$ , are the ultrapowers $(\omega+1)^\omega/{\cal U}_i$ necessarily isomorphic as lattices for $i = 1,2$?
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### Limit of the Schröder numbers ratio

I have been playing around with interesting integer sequences and came across Schröder number which defines the number of lattice paths of n x n grid. The recurrence formula to calculate these numbers ...
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### The Tarski-Lindenbaum theorem of the middle value

In Ettore Casari, La Matematica della Verita' (2006), the following theorem (attributed to Tarski and Lindenbaum 1926), is stated, which Casari calls the theorem of the middle value of Tarski-...
1 vote
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1 vote
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### Automorphism groups of which lattices act irreducibly on the ambient Euclidean space

(I asked this question on MSE a few days ago but it hasn't drawn any response yet.) Let $V$ be a finite-dimensional real inner product space and let $L \subset V$ be a lattice of full rank. Consider ...
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### Covering radius of a lattice from relevant vectors

Let $L$ be an $n-$dimensional lattice. The Voronoi region of $L$ is given by $$\mathcal{V}(L)=\big\{x\in\mathbb{R}^n~|~ \|x\|_2\leq \|x-v\|_2~\forall v\in L\setminus\{0\}\big\}.$$ Considering the ...
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### Why is Schröder numbers equivalent to the number of perfect matchings for triangular grid of n squares and how the graph look like? [duplicate]

In the OEIS entry for the Schröder numbers is A006318. There is a comment which related the sequence to perfect matchings: The number of perfect matchings in a triangular grid of n squares (n = 1, 4, ...
I'm studying a geometrical problem where an (even, negative-definite) lattice $L$ arises. Roughly, as an intersection pairing for curves on a surface. In fact, the problem naturally leads me to ...