Independence Inheritance and Diophantine Approximation for Systems of Linear Forms

Abstract

The classical Khintchine–Groshev theorem is a generalization of Khintchine’s theorem on simultaneous Diophantine approximation, from approximation of points in Rm to approximation of systems of linear forms in Rnm. 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 DuffinSchaeffer 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) × m)-dimensional Khintchine–Groshev theorem (k ≥ 0) are always k-levels more probabilistically independent than the sets involved the (n × m)dimensional theorem. Hence, it is shown that Khintchine’s theorem itself underpins the Khintchine–Groshev theory.