Simultaneous inhomogeneous Diophantine approximation on manifolds

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dc.creatorBeresnevich, Victor
dc.creatorVelani, Sanju
dc.date2007-10-30
dc.date.accessioned2026-07-07T08:39:30Z
dc.date.available2026-07-07T08:39:30Z
dc.descriptionIn 1998, Kleinbock & Margulis established a conjecture of V.G. Sprindzuk in metrical Diophantine approximation (and indeed the stronger Baker-Sprindzuk conjecture). In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent $w_{0}(\vv x) = 1/n$ for almost every point $\vv x$ on a non-degenerate submanifold $\cM$ of $\R^n$. In this paper the simultaneous inhomogeneous analogue of Sprindzuk's conjecture is established. More precisely, for any `inhomogeneous' vector $\bmθ\in\R^n$ we prove that the simultaneous inhomogeneous Diophantine exponent $w_{0}(\vv x, \bmθ)= 1/n$ for almost every point $\vv x$ on $M$. The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent $w_0(\vv x)=1/n$ for almost all $\vv x\in \cM$ if and only if for any $\bmθ\in\R^n$ the inhomogeneous exponent $w_0(\vv x,\bmθ)=1/n$ for almost all $\vv x\in \cM$. The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered in \cite{Beresnevich-Velani-new-inhom}. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas of \cite{Beresnevich-Velani-new-inhom} while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory.
dc.descriptionDedicated to A.O. Gelfond on what would have been his 100th birthday 13 pages
dc.identifierhttps://arxiv.org/abs/0710.5685
dc.identifierhttp://arxiv.org/abs/0710.5685
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/141094
dc.subjectNumber Theory
dc.titleSimultaneous inhomogeneous Diophantine approximation on manifolds
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