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Results tagged with lattices
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user 46140
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])
5
votes
Accepted
The exact number of points within a circle of radius r centered on a lattice point in a hexa...
By identifying the lattice points with numbers of the form $x - y\omega$, $\omega = e^{2\pi i / 3}$, $x, y \in \mathbb{Z}$, we find that we want to count Diophantine solutions to $x^2 + xy + y^2 \le r …
- 7,226
6
votes
How do we know there are no more Deligne–Mostow/Thurston lattices?
Note that Thurston says
Mostow has rigorously enumerated examples by hand, so this table can be regarded as just a check.
That is effectively a claim that it suffices to check up to denominator $42$ …
- 7,226
1
vote
Accepted
On "The Average Height of Planted Plane Trees" by Knuth, de Bruijn and Rice (1972)
I think the problem may be in the bijection. Consider instead the following bijection between (planted plane) trees with $n$ vertices and Dyck paths from $(0,0)$ to $(n-1,n-1)$: perform a depth-first …