%0 Journal Article %J arXiv preprint arXiv:2403.02480 %D 2024 %T Approximating rational points on surfaces %A Brian Lehmann %A David McKinnon %A Matthew Satriano %X

Let X be a smooth projective algebraic variety over a number field k and P in X(k). In 2007, the second author conjectured that, in a precise sense, if rational points on X are dense enough, then the best rational approximations to P must lie on a curve. We present a strategy for deducing a slightly weaker conjecture from Vojta's conjecture, and execute this strategy for the full conjecture for split surfaces.

%B arXiv preprint arXiv:2403.02480 %G eng %U http://arxiv.org/abs/2403.02480 %0 Journal Article %D 2023 %T ERRATUM FOR APPROXIMATING RATIONAL POINTS ON TORIC VARIETIES %A McKinnon, David %A Satriano, Matthew %G eng %0 Journal Article %J Algebraic Geometry (to appear) %D 2023 %T Rational approximation on cubic hypersurfaces %A McKinnon, David %B Algebraic Geometry (to appear) %G eng %U arXiv preprint arXiv:2310.01640 %0 Journal Article %J J. Algebraic Geometry %D 2022 %T Codimension two integral points on some rationally connected threefolds are potentially dense %A McKinnon, David %A Roth, Mike %B J. Algebraic Geometry %V 31 %P 345-386 %G eng %0 Journal Article %J Transactions of the American Mathematical Society %D 2021 %T Approximating rational points on toric varieties %A McKinnon, David %A Satriano, Matthew %B Transactions of the American Mathematical Society %V 374 %P 3557–3577 %G eng %0 Journal Article %J NEW YORK JOURNAL OF MATHEMATICS %D 2021 %T On curves with high multiplicity on P (a, b, c) for min (a, b, c) %A McKinnon, David %A Razafy, Rindra %A Satriano, Matthew %A Sun, Yuxuan %B NEW YORK JOURNAL OF MATHEMATICS %I ELECTRONIC JOURNALS PROJECT UNIV ALBANY, DEPT MATHEMATICS & SCIENCE, ALBANY … %V 27 %P 1060–1084 %G eng %0 Journal Article %J Linear Algebra and its Applications %D 2021 %T What fraction of an Sn-orbit can lie on a hyperplane? %A Huang, Jiahui %A McKinnon, David %A Satriano, Matthew %B Linear Algebra and its Applications %I North-Holland %V 613 %P 1–23 %G eng %0 Unpublished Work %D 2019 %T Counting rational points on ruled varieties over function fields %A McKinnon, David %G eng %0 Journal Article %J arXiv preprint arXiv:1806.03180 %D 2018 %T The arithmetic puncturing problem and integral points %A McKinnon, David %A Zhu, Yi %B arXiv preprint arXiv:1806.03180 %G eng %0 Journal Article %J European Journal of Mathematics %D 2016 %T An analogue of Liouville’s theorem and an application to cubic surfaces %A McKinnon, David %A Roth, Mike %B European Journal of Mathematics %I Springer International Publishing Cham %V 2 %P 929–959 %G eng %0 Online Database %D 2016 %T PDF file containing tables of generators of various cones in the Néron-Severi group of a smooth cubic surface %A David McKinnon %G eng %U https://www.math.uwaterloo.ca/~dmckinno/Papers/cubictable.pdf %0 Journal Article %J Journal of Mathematical Physics %D 2015 %T Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets %A Forest, Simon %A Gosset, David %A Kliuchnikov, Vadym %A McKinnon, David %B Journal of Mathematical Physics %I AIP Publishing %V 56 %G eng %0 Journal Article %J Inventiones mathematicae %D 2015 %T Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties %A McKinnon, David %A Roth, Mike %B Inventiones mathematicae %I Springer Berlin Heidelberg %V 200 %P 513–583 %G eng %0 Journal Article %J Algebra & Number Theory %D 2012 %T Ideals of degree one contribute most of the height %A Levin, Aaron %A McKinnon, David %B Algebra & Number Theory %I Mathematical Sciences Publishers %V 6 %P 1223–1238 %G eng %0 Journal Article %J Bulletin of the London Mathematical Society %D 2011 %T Vojta's conjecture implies the Batyrev–Manin conjecture for K 3 surfaces %A McKinnon, David %B Bulletin of the London Mathematical Society %I Oxford University Press %V 43 %P 1111–1118 %G eng %0 Journal Article %J Algebra & Number Theory %D 2010 %T Density of rational points on diagonal quartic surfaces %A Logan, Adam %A McKinnon, David %A van Luijk, Ronald %B Algebra & Number Theory %I Mathematical Sciences Publishers %V 4 %P 1–20 %G eng %0 Journal Article %J Journal of Number Theory %D 2010 %T K3 surfaces, rational curves, and rational points %A Baragar, Arthur %A McKinnon, David %B Journal of Number Theory %I Academic Press %V 130 %P 1470–1479 %G eng %0 Journal Article %J Journal de théorie des nombres de Bordeaux %D 2009 %T Patterns and periodicity in a family of resultants %A Hare, Kevin G %A McKinnon, David %A Sinclair, Christopher D %B Journal de théorie des nombres de Bordeaux %V 21 %P 215–234 %G eng %0 Journal Article %J The Quarterly Journal of Mathematics %D 2008 %T On the error terms and exceptional sets in conjectural second main theorems %A Levin, Aaron %A McKinnon, David %A Winkelmann, Jörg %B The Quarterly Journal of Mathematics %I Oxford University Press %V 59 %P 487–498 %G eng %0 Journal Article %J Journal of Graph Theory %D 2008 %T Straight line embeddings of cubic planar graphs with integer edge lengths %A Geelen, Jim %A Guo, Anjie %A McKinnon, David %B Journal of Graph Theory %I Wiley Subscription Services, Inc., A Wiley Company Hoboken %V 58 %P 270–274 %G eng %0 Journal Article %J Journal of Algebraic Geometry %D 2007 %T A conjecture on rational approximations to rational points %A David McKinnon %B Journal of Algebraic Geometry %V 16 %P 257-303 %G eng %0 Journal Article %J The Rocky Mountain Journal of Mathematics %D 2007 %T Curves arising from endomorphism rings of Kronecker modules %A McKinnon, David %A Roth, Mike %B The Rocky Mountain Journal of Mathematics %I The Rocky Mountain Mathematics Consortium %P 879–892 %G eng %0 Journal Article %J The American Mathematical Monthly %D 2005 %T Gauss's lemma for number fields %A Magidin, Arturo %A McKinnon, David %B The American Mathematical Monthly %I Taylor & Francis %V 112 %P 385–416 %G eng %0 Journal Article %J Transactions of the American Mathematical Society %D 2005 %T Lagrangian tori in homotopy elliptic surfaces %A Etgü, Tolga %A McKinnon, David %A Park, B %B Transactions of the American Mathematical Society %V 357 %P 3757–3774 %G eng %0 Journal Article %J Canadian Mathematical Bulletin %D 2004 %T Counting rational points on ruled varieties %A McKinnon, David %B Canadian Mathematical Bulletin %I Cambridge University Press %V 47 %P 264–270 %G eng %0 Journal Article %J Canadian Mathematical Bulletin %D 2004 %T A reduction of the Batyrev-Manin conjecture for Kummer surfaces %A McKinnon, David %B Canadian Mathematical Bulletin %I Cambridge University Press %V 47 %P 398–406 %G eng %0 Journal Article %J Proceedings of the American Mathematical Society %D 2003 %T Vojta’s main conjecture for blowup surfaces %A McKinnon, David %B Proceedings of the American Mathematical Society %V 131 %P 1–12 %G eng %0 Journal Article %J Compositio Mathematica %D 2001 %T An arithmetic analogue of Bezout's theorem %A McKinnon, David %B Compositio Mathematica %I London Mathematical Society %V 126 %P 147–155 %G eng %0 Journal Article %J Journal of Number Theory %D 2000 %T Counting rational points on K3 surfaces %A McKinnon, David %B Journal of Number Theory %I Academic Press %V 84 %P 49–62 %G eng %0 Thesis %D 1999 %T Counting rational points of bounded height on algebraic varieties %A McKinnon, David Kerr %I University of California, Berkeley %G eng %9 PhD thesis