Diophantine approximation on curves and the distribution of rational points: Contributions to the divergence theory

Abstract In this paper we develop an explicit method for studying the distribution of rational points near manifolds. As a consequence we obtain optimal lower bounds on the number of rational points of bounded height lying at a given distance from an arbitrary non-degenerate curve in R n . This generalises previous results for analytic non-degenerate curves. Furthermore, the main results are proved in the inhomogeneous setting. For n ≥ 3 , the inhomogeneous aspect is new even under the additional assumption of analyticity. Applications of the main distribution theorem also include the inhomogeneous Khintchine-Jarnik type theorem for divergence for arbitrary non-degenerate curves in R n .

[1]  Diophantine approximation on planar curves: the convergence theory , 2006, math/0602225.

[2]  Jing-Jing Huang.  The density of rational points near hypersurfaces , 2017, 1711.01390.

[3]  V. Jarník.  Über die simultanen diophantischen Approximationen , 1931 .

[4]  V. Beresnevich.  Rational points near manifolds and metric Diophantine approximation , 2009, 0904.0474.

[5]  Victor Beresnevich,et al.  A Groshev Type Theorem for Convergence on Manifolds , 2002 .

[6]  R. Vaughan,et al.  Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory , 2015, 1506.09049.

[7]  V. Beresnevich.  Badly approximable points on manifolds , 2013, 1304.0571.

[8]  R. Vaughan,et al.  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) , 2007 .

[9]  Jing-Jing Huang.  Rational points near planar curves and Diophantine approximation , 2014, 1403.7388.

[10]  Dmitry Kleinbock,et al.  Flows on homogeneous spaces and Diophantine approximation on manifolds , 1998, math/9810036.

[11]  V. Beresnevich,et al.  Explicit bounds for rational points near planar curves and metric Diophantine approximation , 2010, 1002.2803.

[12]  Victor Beresnevich,et al.  A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures , 2004 .

[13]  Wolfgang M. Schmidt,et al.  Diophantine Approximations and Diophantine Equations , 1991 .

[14]  Victor Beresnevich,et al.  Measure Theoretic Laws for Lim Sup Sets , 2004 .

[15]  DIOPHANTINE APPROXIMATION ON MANIFOLDS AND LOWER BOUNDS FOR HAUSDORFF DIMENSION , 2017, 1712.03761.

[16]  A. Khintchine,et al.  Zur metrischen Theorie der diophantischen Approximationen , 1926 .

[17]  David Simmons.  Some manifolds of Khinchin type for convergence , 2016, 1602.01727.

[18]  Counting rational points near planar curves , 2014, 1401.4958.

[19]  Victor Beresnevich,et al.  Metric Diophantine Approximation : aspects of recent work , 2016, 1601.01948.

[20]  G. A. Margulis,et al.  Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions , 2001 .

[21]  V. Beresnevich,et al.  Inhomogeneous Diophantine approximation on planar curves , 2009, 0903.2817.