On estimates for integral solutions of linear inequalities (Q1057916)

\textit{E. Bombieri} and \textit{J. Vaaler} [Invent. Math. 73, 11--32 (1983), Addendum 75, 377 (1984; Zbl 0533.10030)] obtained an adelic generalization of Minkowski's theorem on successive minima in geometry of numbers and derived an important effective formulation of the well-known Siegel lemma on the size of integral solutions of linear equations. In a similar context involving linear inequalities this paper is concerned with an analogue of a theorem of \textit{A. Khintchine} [Izv. Akad. Nauk SSSR, Ser. Mat. 12, 113--122 (1948; Zbl 0030.02002)] on integral solutions for inequalities arising from systems of linear forms and also with an analogue of a Kronecker-type theorem with regard to Euclidean frames of integral vectors. The proof of the former theorem invokes Bombieri-Vaaler's adelic generalization of Minkowski's theorem on successive minima. [For Minkowski's theorem on successive minima over the adèles, see also \textit{R. B. McFeat}, Diss. Math. 88 (1971; Zbl 0229.10014).]