Expand all Collapse all | Results 1 - 3 of 3 |
1. CJM 1999 (vol 51 pp. 1258)
Similarity Submodules and Root Systems in Four Dimensions Lattices and $\ZZ$-modules in Euclidean space possess an infinitude
of subsets that are images of the original set under similarity
transformation. We classify such self-similar images according to
their indices for certain 4D examples that are related to 4D root
systems, both crystallographic and non-crystallographic. We
encapsulate their statistics in terms of Dirichlet series
generating functions and derive some of their asymptotic properties.
Categories:11S45, 11H05, 52C07 |
2. CJM 1999 (vol 51 pp. 1300)
On the Existence of Similar Sublattices Partial answers are given to two questions. When does a lattice
$\Lambda$ contain a sublattice $\Lambda'$ of index $N$ that is
geometrically similar to $\Lambda$? When is the sublattice
``clean'', in the sense that the boundaries of the Voronoi cells
for $\Lambda'$ do not intersect $\Lambda$?
Category:52C07 |
3. CJM 1999 (vol 51 pp. 225)
Asymptotic Formulae for the Lattice Point Enumerator Let $M$ be a convex body such that the boundary has positive
curvature. Then by a well developed theory dating back to Landau and
Hlawka for large $\lambda$ the number of lattice points in $\lambda M$
is given by $G(\lambda M) =V(\lambda M) + O(\lambda^{d-1-\varepsilon
(d)})$ for some positive $\varepsilon(d)$. Here we give for general
convex bodies the weaker estimate
\[
\left| G(\lambda M) -V(\lambda M) \right |
\le \frac{1}{2} S_{\Z^d}(M) \lambda^{d-1}+o(\lambda^{d-1})
\]
where $S_{\Z^d}(M)$ denotes the lattice surface area of $M$. The term
$S_{\Z^d}(M)$ is optimal for all convex bodies and $o(\lambda^{d-1})$
cannot be improved in general. We prove that the same estimate even
holds if we allow small deformations of $M$.
Further we deal with families $\{P_\lambda\}$ of convex bodies where
the only condition is that the inradius tends to infinity. Here we have
\[
\left| G(P_\lambda)-V(P_\lambda) \right|
\le dV(P_\lambda,K;1)+o \bigl( S(P_\lambda) \bigr)
\]
where the convex body $K$ satisfies some simple condition,
$V(P_\lambda,K;1)$ is some mixed volume and $S(P_\lambda)$ is the
surface area of $P_\lambda$.
Categories:11P21, 52C07 |