Independence Inheritance and Diophantine Approximation for Systems of Linear Forms

Abstract

Abstract The classical Khintchine–Groshev theorem is a generalization of Khintchine’s theorem on simultaneous Diophantine approximation, from approximation of points in ${\mathbb {R}}^m$ to approximation of systems of linear forms in ${\mathbb {R}}^{nm}$. In this paper, we present an inhomogeneous version of the Khintchine–Groshev theorem that does not carry a monotonicity assumption when $nm>2$. Our results bring the inhomogeneous theory almost in line with the homogeneous theory, where it is known by a result of Beresnevich and Velani [11] that monotonicity is not required when $nm>1$. That result resolved a conjecture of Beresneich et al. [5], and our work resolves almost every case of the natural inhomogeneous generalization of that conjecture. Regarding the two cases where $nm=2$, we are able to remove monotonicity by assuming extra divergence of a measure sum, akin to a linear forms version of the Duffin–Schaeffer conjecture. When $nm=1$, it is known by work of Duffin and Schaeffer [16] that the monotonicity assumption cannot be dropped. The key new result is an independence inheritance phenomenon; the underlying idea is that the sets involved in the $((n+k)\times m)$-dimensional Khintchine–Groshev theorem ($k\geq 0$) are always $k$-levels more probabilistically independent than the sets involved the $(n\times m)$-dimensional theorem. Hence, it is shown that Khintchine’s theorem itself underpins the Khintchine–Groshev theory.