Inhomogeneous Diophantine approximation on planar curves

V. V. Beresnevich; R. C. Vaughan; S. L. Velani

doi:10.1007/s00208-010-0548-9

Inhomogeneous Diophantine approximation on planar curves

V. V. Beresnevich, R. C. Vaughan, S. L. Velani

Mathematics

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18 Scopus citations

Abstract

In this paper we develop the inhomogeneous metric theory of simultaneous Diophantine approximation on planar curves. Our results naturally extend the homogeneous Khintchine and Jarník type theorems established in Beresnevich et al. (Ann Math 166(2):367-426, 2007) and Vaughan and Velani (Invent Math 166:103-124, 2006) and are the first of their kind. The key lies in obtaining essentially the best possible results regarding the distribution of 'shifted' rational points near planar curves.

Original language	English (US)
Pages (from-to)	929-942
Number of pages	14
Journal	Mathematische Annalen
Volume	349
Issue number	4
DOIs	https://doi.org/10.1007/s00208-010-0548-9
State	Published - Apr 2011

All Science Journal Classification (ASJC) codes

Mathematics(all)

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title = "Inhomogeneous Diophantine approximation on planar curves",

abstract = "In this paper we develop the inhomogeneous metric theory of simultaneous Diophantine approximation on planar curves. Our results naturally extend the homogeneous Khintchine and Jarn{\'i}k type theorems established in Beresnevich et al. (Ann Math 166(2):367-426, 2007) and Vaughan and Velani (Invent Math 166:103-124, 2006) and are the first of their kind. The key lies in obtaining essentially the best possible results regarding the distribution of 'shifted' rational points near planar curves.",

author = "Beresnevich, {V. V.} and Vaughan, {R. C.} and Velani, {S. L.}",

note = "Funding Information: V.V. Beresnevich: EPSRC Advanced Research Fellow, grant EP/C54076X/1; R.C. Vaughan: Research supported in part by NSA grant MDA904-03-1-0082 and by a Distinguished Visitor Fellowship at the University of York; S.L. Velani: Research supported by EPSRC grants EP/E061613/1 and EP/F027028/1.",

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AB - In this paper we develop the inhomogeneous metric theory of simultaneous Diophantine approximation on planar curves. Our results naturally extend the homogeneous Khintchine and Jarník type theorems established in Beresnevich et al. (Ann Math 166(2):367-426, 2007) and Vaughan and Velani (Invent Math 166:103-124, 2006) and are the first of their kind. The key lies in obtaining essentially the best possible results regarding the distribution of 'shifted' rational points near planar curves.

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Inhomogeneous Diophantine approximation on planar curves

Abstract

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