Independence inheritance and Diophantine approximation for systems of linear forms

TL;DR

This paper extends the inhomogeneous Khintchine-Groshev theorem to cases without monotonicity assumptions for systems of linear forms, revealing an independence inheritance phenomenon that links higher-dimensional cases to the classical theorem.

Contribution

It introduces an independence inheritance principle that connects higher-dimensional Diophantine approximation results to the classical Khintchine-Groshev theorem, removing monotonicity constraints in many cases.

Findings

Inhomogeneous Khintchine-Groshev theorem extended without monotonicity for nm>2.

Independence inheritance phenomenon links higher-dimensional sets to classical results.

Almost all cases of the inhomogeneous conjecture are resolved, except for nm=2 with extra divergence conditions.

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 which 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 (2010) that monotonicity is not required when $nm>1$. That result resolved a conjecture of Beresnevich, Bernik, Dodson, and Velani (2009), 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 (1941) 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.