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1. 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 |