The metrical theory of simultaneously small linear forms

Abstract

In this paper we investigate the metrical structure of the set of all points $X\in\mathbb{R}^n$ which satisfy a simultaneously small system of Diophantine inequalities for infinitely many integer vectors. We establish the complete metric theory for the given system which implies a general Khintchine--Groshev type theorem, as well as its Hausdorff measure generalization. The latter includes the original dimension results obtained in [5] as special cases.