Asymptotic Formulae for the Lattice Point Enumerator

Betke, U. ; Jr., K. Böröczky

Canadian journal of mathematics, Tome 51 (1999) no. 2, pp. 225-249

Résumé

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 $G\left( \lambda M \right)=V\left( \lambda M \right)+O\left( {{\lambda }^{d-1-\varepsilon \left( d \right)}} \right)$ for some positive $\varepsilon (d)$ . Here we give for general convex bodies the weaker estimate $$|G\left( \lambda M \right)-V\left( \lambda M \right)|\,\le \,\frac{1}{2}{{S}_{{{Z}^{d}}}}\left( M \right){{\lambda }^{d-1}}+o\left( {{\lambda }^{d-1}} \right)$$ where ${{S}_{{{Z}^{d}}}}\left( M \right)$ denotes the lattice surface area of $M$ . The term ${{S}_{{{Z}^{d}}}}\left( M \right)$ is optimal for all convex bodies and $o\left( {{\lambda }^{d-1}} \right)$ 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 $\left\{ {{P}_{\lambda }} \right\}$ of convex bodies where the only condition is that the inradius tends to infinity. Here we have $$|G\left( {{P}_{\lambda }} \right)-V\left( {{P}_{\lambda }} \right)|\,\le \,dV\left( {{P}_{\lambda }},\,K;1 \right)+o\left( S\left( {{P}_{\lambda }} \right) \right)$$ where the convex body $K$ satisfies some simple condition, $V\left( {{P}_{\lambda }},K;1 \right)$ is some mixed volume and $S\left( {{P}_{\lambda }} \right)$ is the surface area of ${{P}_{\lambda }}$ .

Détail

DOI : 10.4153/CJM-1999-012-9

Mots-clés : 11P21, 52C07

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