Mahler, Kurt

Reprint: On lattice points in \(n\)-dimensional star bodies (1946)

Doc. Math. Extra Vol. Mahler Selecta, 483-520 (2019)
DOI: 10.25537/dm.2019.SB-483-520


Let \(F(X)=F(x_1,\ldots,x_n)\) be a continuous non-negative function of \(X=(x_1,\ldots,x_n)\) that satisfies \(F(tX)=|t|F(X)\) for all real numbers \(t\). The set \(K\) in \(n\)-dimensional Euclidean space \(\mathbb{R}^n\) defined by \(F(X)\leqslant 1\) is called a star body. In this paper, Mahler studies the lattices \(\Lambda\) in \(\mathbb{R}^n\) which are of minimum determinant and have no point except \((0,\ldots,0)\) inside \(K\). He investigates how many points of such lattices lie on, or near to, the boundary of \(K\), and considers in detail the case when \(K\) admits an infinite group of linear transformations into itself. \par Reprint of the author's paper [Proc. R. Soc. Lond., Ser. A 187, 151--187 (1946; Zbl 0060.11710)].

Mathematics Subject Classification

11-03, 11H16