On estimates for integral solutions of linear inequalities

Abstract

Recently, Bombieri and Vaaler obtained an interesting adelic formulation of the first and the second theorems of Minkowski in the Geometry of Numbers and derived an effective formulation of the well-known "Siegel's 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 Khintchine 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 formulation of Minkowski's theorem.