Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents

Abstract

We extend the Khintchine transference inequalities, as well as a homogeneous-inhomogeneous transference inequality for lattices, due to Bugeaud and Laurent, to a weighted setting. We also provide applications to inhomogeneous Diophantine approximation on manifolds and to weighted badly approximable vectors. Finally, we interpret and prove a conjecture of Beresnevich-Velani (2010) about inhomogeneous intermediate exponents.2010 Mathematics Subject Classification. 11J83.

Dirichlet is not just Bad and Singular

“…, y n , 1) : y ∈ L} of L into R n+1 . This notion of height is relatively standard and is usually referred to as the projective or Weil height of L -see [7,17,21] for more details. Note that when d = 0, L corresponds to a rational point p q := p 1 q , .…”

A graph arising in the Geometry of Numbers

“…The regular graph of [6] inspired a conjecture on diophantine approximation which was established in [3], [4]. As recently more attention has been devoted to diophantine approximation with weights (see for example [1], [2], [7], [8]), it may be expected that the present graph will have application in the weighted setting.…”

“…We restrict the picture to the part with q in the interval [τ 0 , τ 1 ] = [1,4]. It is plain to see that this graph is proper as any b 0 r = σ r (1, v r ) lies above the corresponding a 0 r = σ r (1, u r ), so that v r > u r .…”

Singular vectors on manifolds and fractals

“…Exact transference theorems for the weighted setting were obtained quite recently. Improving on a paper by Chow, Ghosh, Guan, Marnat and Simmons [9], German [15] proved a transference inequality which in particular states that…”