Authors: V V Beresnevich S L Velani
Publish Date: 2012/01/07
Volume: 180, Issue: 5, Pages: 531-541
Abstract
In 1998 Kleinbock and Margulis proved Sprindzuk’s conjecture pertaining to metrical Diophantine approximation and indeed the stronger Baker–Sprindzuk conjecture In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent w 0x = 1/n for almost every point x on a nondegenerate submanifold mathcalM of mathbbRn In this paper the simultaneous inhomogeneous analogue of Sprindzuk’s conjecture is established More precisely for any “inhomogeneous” vector θ ∈ mathbbRn we prove that the simultaneous inhomogeneous Diophantine exponent w 0x θ is 1/n for almost every point x on mathcalM The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent w 0x is 1/n for almost all x ∈ mathcalM if and only if for any θ ∈ mathbbRn the inhomogeneous exponent w 0x θ = 1/n for almost all x ∈ mathcalM The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered by us Nevertheless it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory
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