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تسجيل مستخدم جديدDistribution of Angles in Hyperbolic Lattices
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We prove an effective equidistribution result about angles in a hyperbolic lattice. We use this to generalize a result due to F. P. Boca.
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It is well known that the angles in a lattice acting on hyperbolic $n$-space become equidistributed. In this paper we determine a formula for the pair correlation density for angles in such hyperbolic lattices. Using this formula we determine, among
other things, the asymptotic behavior of the density function in both the small and large variable limits. This extends earlier results by Boca, Pasol, Popa and Zaharescu and Kelmer and Kontorovich in dimension 2 to general dimension $n$. Our proofs use the decay of matrix coefficients together with a number of careful estimates, and lead to effective results with explicit rates.
We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixe
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is uniformly distributed modulo one. We reduce the problem to an equidistribution theorem of Anton Good concerning the orbits of a point in the upper half-plane under the action of a Fuchsian group.
We explore whether a root lattice may be similar to the lattice $mathscr O$ of integers of a number field $K$ endowed with the inner product $(x, y):={rm Trace}_{K/mathbb Q}(xcdottheta(y))$, where $theta$ is an involution of $K$. We classify all pair
s $K$, $theta$ such that $mathscr O$ is similar to either an even root lattice or the root lattice $mathbb Z^{[K:mathbb Q]}$. We also classify all pairs $K$, $theta$ such that $mathscr O$ is a root lattice. In addition to this, we show that $mathscr O$ is never similar to a positive-definite even unimodular lattice of rank $leqslant 48$, in particular, $mathscr O$ is not similar to the Leech lattice. In appendix, we give a general cyclicity criterion for the primary components of the discriminant group of $mathscr O$.
For a positive integer $s$, a lattice $L$ is said to be $s$-integrable if $sqrt{s}cdot L$ is isometric to a sublattice of $mathbb{Z}^n$ for some integer $n$. Conway and Sloane found two minimal non $2$-integrable lattices of rank $12$ and determinant
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