View-obstruction problems and Kronecker's theorem. II
[Part I, cf. Proc. Am. Math. Soc. 123, No. 11, 3279-3284 (1995; Zbl 0855.11034).] One of the main results of this paper is as follows: Let \(\alpha_1,\dots,\alpha_n\) be real numbers and \(M\geq 2\). If \((a_1,\dots,a_n)\in \mathbb{R}^n\) such that there exists no nonzero tuple \((c_1,\dots,c_n)\in \mathbb{Z}^n\) with \(|c_j|\leq M\) \((1\leq j\leq n)\) satisfying \(c_1a_1+\dots+ c_na_n=0\), then \[ \inf_{t\in\mathbb{R}} \sum^n_{i=1} |a_i t-\alpha_i|^2< 2n/3([M]+ 1/8)^2, \] where \(|x|\) is the distance from \(x\) to the nearest integer, \([a]\) means the integral part of the real number \(a\). It is a new quantitative form of Kronecker's theorem and improves the result of \textit{H. M. Bacon} [Ann. Math., II. Ser. 35, 776-786 (1934; Zbl 0010.15205)]. Using it the author improves some results on \(\lambda(n)\) (resp. \(\nu(n)\)) in the view-obstruction problem for cubes (resp. spheres) in \(E_n\) proposed by \textit{T. W. Cusick} [Aequat. Math. 9, 165-170 (1973; Zbl 0265.52003)]. Furthermore, the author introduces and studies the sup-inf value \[ K_F=K_F(\alpha_1,\dots, \alpha_n)= \sup_{a_i\in\mathbb{R}_+} \inf_{t\in\mathbb{R}} F(|a_1t-\alpha_1|, \dots,|a_nt-\alpha_n|) \] for a function \(F\) defined on \([0,1/2]^n\) and a tuple \((\alpha_1,\dots,\alpha_n)\in \mathbb{R}^n\), which is a generalization of \(\lambda(n)\) and \(\nu(n)\).
scientific article; zbMATH DE number 908252