1. CJM 1999 (vol 51 pp. 1258)
 Baake, Michael; Moody, Robert V.

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 selfsimilar images according to
their indices for certain 4D examples that are related to 4D root
systems, both crystallographic and noncrystallographic. 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)
3. CJM 1999 (vol 51 pp. 225)
 Betke, U.; Böröczky, K.

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 $G(\lambda M) =V(\lambda M) + O(\lambda^{d1\varepsilon
(d)})$ for some positive $\varepsilon(d)$. Here we give for general
convex bodies the weaker estimate
\[
\left G(\lambda M) V(\lambda M) \right 
\le \frac{1}{2} S_{\Z^d}(M) \lambda^{d1}+o(\lambda^{d1})
\]
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 $o(\lambda^{d1})$
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 
