tailieunhanh - Đề tài " Diophantine approximation on planar curves and the distribution of rational points "

Sums of two squares near perfect squares by R. C. Vaughan∗∗∗ In memory of Pritish Limani (1983–2003) Abstract Let C be a nondegenerate planar curve and for a real, positive decreasing function ψ let C(ψ) denote the set of simultaneously ψ-approximable points lying on C. We show that C is of Khintchine type for divergence; . if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(ψ) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. . | Annals of Mathematics Diophantine approximation on planar curves and the distribution of rational points By Victor Beresnevich Detta Dickinson and Sanju Velanin Annals of Mathematics 166 2007 367 426 Diophantine approximation on planar curves and the distribution of rational points By Victor Beresnevich Detta Dickinson and Sanju Velani With an appendix Sums of two squares near perfect squares by R. C. Vaughan In memory of Pritish Limani 1983-2003 Abstract Let C be a nondegenerate planar curve and for a real positive decreasing function 0 let C 0 denote the set of simultaneously 0-approximable points lying on C. We show that C is of Khintchine type for divergence . if a certain sum diverges then the one-dimensional Lebesgue measure on C of C 0 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 0 with lower order in a critical range we determine a general exact formula for the Hausdorff dimension of C 0 . These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds. Contents 1. Introduction . Background and the general problem . The Khintchine type theory . The Khintchine theory for rational quandrics . The Hausdorff measure dimension theory . Rational points close to a curve This work has been partially supported by INTAS Project 00-429 and by EPSRC grant GR R90727 01. Royal Society University Research Fellow. Research supported by NSA grant MDA904-03-1-0082. 368 VICTOR BERESNEVICH DETTA DICKINSON AND SANJU VELANI 2. Proof of the rational quadric statements . Proof of Theorem 2 . Hausdorff measure and dimension . Proof of Theorem 5 3. Ubiquitous systems . Ubiquitous systems in R . Ubiquitous systems close to a curve in R 4. Proof of Theorem 6 . The ubiquity version of Theorem 6