Asymptotic directions for best approximations of \(n\)-dimensional linear forms
For \(\alpha \) in Euclidean \(n\)-space let \({\mathcal L}_{\alpha }: x \mapsto (x,\alpha )\) denote the linear form induced by \(\alpha \). An integral vector \(m\in {\mathbb Z}^n\) is called a \textit{best approximation} (resp. \textit{one-sided best approximation}) of \({\mathcal L}_{\alpha }\), if \(L_{\alpha }(m)\) minimizes the distance to \({\mathbb Z}\) (resp. the fractional part) on the set of integral vectors of smaller norm. These definitions depend on the choice of an arbitrary fixed norm determined by a convex radially symmetric function. The paper investigates the asymptotic behaviour of the one-sided best approximations.
asymptotic directions of best approximations
scientific article; zbMATH DE number 5132499