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 R m to approximation of systems of linear forms in 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) × 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.

Metric results on inhomogeneously singular vectors

“…This shows the left inequality in (35). Similarly for odd j with p 0,j = ⌊b h j x 0 ⌋ we have 1) . This shows the right inequality in (35).…”

“…be any norm on R m . By Dirichlet's Theorem, for any ξ ∈ R m and any Q ≥ 1 the homogeneous system of inequalities (1) 1…”

“…Fixing θ ∈ R m , from a metrical point of view, for ordinary approximation we still have a very satisfactory understanding by the inhomogeneous Khintchine-Groshev Theorem, analogous to the homogeneous theory. See [1] for recent subtle improvements and further references. On the other hand, little is known when extending Problem 1 and related metrical questions on uniform approximation to the inhomogeneous setting, not even for the maybe most interesting value w = 1/m.…”