Abstract
Let \({\mathcal {C}}\) be two times continuously differentiable curve in \({\mathbb {R}}^2\) with at least one point at which the curvature is non-zero. For any \(i,j \geqslant 0\) with \(i+j =1\), let \({\mathbf {Bad}}(i,j)\) denote the set of points \((x,y) \in {\mathbb {R}}^2\) for which \( \max \{ \Vert qx\Vert ^{1/i}, \, \Vert qy\Vert ^{1/j} \} > c/q \) for all \( q \in {\mathbb {N}}\). Here \(c = c(x,y)\) is a positive constant. Our main result implies that any finite intersection of such sets with \({\mathcal {C}}\) has full Hausdorff dimension. This provides a solution to a problem of Davenport dating back to the sixties.
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Notes
Added in proof: An, Beresnevich and the second author have recently proved this winning statement. In fact, winning within the more general inhomogeneous setup is established. A manuscript entitled ‘Badly approximable points on planar curves and winning’ is in preparation.
A few days before completing this paper, Victor Beresnevich communicated to us that he has established Conjecture A under the extra assumption involving the natural analogue of (3). In turn, under this assumption, by making use of Pyartly’s technique he has proved Conjecture C for non-degenerate analytic manifolds. This in our opinion represents a magnificent achievement—see Remark 5.
Added in proof: V. Beresnevich: Badly approximable points on manifolds. Pre-print: arXiv:1304.0571.
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Acknowledgments
We would like to thank the referee for carefully reading the paper and pointing out various inconsistencies. SV would like to thank Haleh Afshar and Maurice Dodson for their fantastic support over the last two decades and for introducing him to Persian rice and Diophantine approximation. What more could a boy possibly want? SV would also like to thank the one and only Bridget Bennett for putting up with his baldness and middle age spread. Finally, much love to those fab girls Ayesha and Iona as they move swiftly into their second decade!
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Dedicated to our mathematical grandparents: Harold Davenport and Maurice Dodson.
S. Velani’s research was partially supported by EPSRC grants EP/E061613/1 and EP/F027028/1.
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Badziahin, D., Velani, S. Badly approximable points on planar curves and a problem of Davenport. Math. Ann. 359, 969–1023 (2014). https://doi.org/10.1007/s00208-014-1020-z
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DOI: https://doi.org/10.1007/s00208-014-1020-z