Diophantine approximation on planar curves and the distribution of rational points (with an appendix "Sum of two squares near perfect squares" by R.C. Vaughan)

V. Beresnevich; Detta Dickinson; S.L. Velani; Robert Vaughan FRS

doi:10.4007/annals.2007.166.367

Diophantine approximation on planar curves and the distribution of rational points (with an appendix "Sum of two squares near perfect squares" by R.C. Vaughan)

V. Beresnevich, Detta Dickinson, S.L. Velani, Robert Vaughan FRS

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let C be a nondegenerate planar curve and for a real, positive decreasing function ¿ let C(¿) denote the set of simultaneously ¿-approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(¿) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that C is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions ¿ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of C(¿). These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.

Original language	English
Pages (from-to)	367-426
Number of pages	59
Journal	Annals of Mathematics
Volume	166
Issue number	2
DOIs	https://doi.org/10.4007/annals.2007.166.367
Publication status	Published - Sept 2007

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10.4007/annals.2007.166.367

http://annals.math.princeton.edu/2007/166-2/p03

Cite this

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title = "Diophantine approximation on planar curves and the distribution of rational points (with an appendix {"}Sum of two squares near perfect squares{"} by R.C. Vaughan)",

abstract = "Let C be a nondegenerate planar curve and for a real, positive decreasing function ¿ let C(¿) denote the set of simultaneously ¿-approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(¿) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that C is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions ¿ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of C(¿). These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.",

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T1 - Diophantine approximation on planar curves and the distribution of rational points (with an appendix "Sum of two squares near perfect squares" by R.C. Vaughan)

AU - Beresnevich, V.

AU - Dickinson, Detta

AU - Velani, S.L.

AU - Vaughan FRS, Robert

PY - 2007/9

Y1 - 2007/9

N2 - Let C be a nondegenerate planar curve and for a real, positive decreasing function ¿ let C(¿) denote the set of simultaneously ¿-approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(¿) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that C is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions ¿ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of C(¿). These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.

AB - Let C be a nondegenerate planar curve and for a real, positive decreasing function ¿ let C(¿) denote the set of simultaneously ¿-approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(¿) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that C is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions ¿ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of C(¿). These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.

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DO - 10.4007/annals.2007.166.367

M3 - Article

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JF - Annals of Mathematics

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