JournalsjemsVol. 24, No. 3pp. 1045–1101

Intrinsic Diophantine approximation on quadric hypersurfaces

  • Lior Fishman

    University of North Texas, Denton, USA
  • Dmitry Kleinbock

    Brandeis University, Waltham, USA
  • Keith Merrill

    Brandeis University, Waltham, USA
  • David Simmons

    University of York, Heslington, UK
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We consider the question of how well points in a quadric hypersurface MRd{M\subseteq\mathbb{R}^d} can be approximated by rational points of QdM{\mathbb{Q}^d\cap M}. This contrasts with the more common setup of approximating points in a manifold by all rational points in Qd{\mathbb{Q}^d}. We provide complete answers to major questions of Diophantine approximation in this context. Of particular interest are the impact of the real and rational ranks of the defining quadratic form, quantities whose roles in Diophantine approximation have never been previously elucidated. Our methods include a correspondence between the intrinsic Diophantine approximation theory on a rational quadric hypersurface and the dynamics of the group of projective transformations which preserve that hypersurface, similar to earlier results in the non-intrinsic setting due to Dani (1986) and Kleinbock–Margulis (1999).

Cite this article

Lior Fishman, Dmitry Kleinbock, Keith Merrill, David Simmons, Intrinsic Diophantine approximation on quadric hypersurfaces. J. Eur. Math. Soc. 24 (2022), no. 3, pp. 1045–1101

DOI 10.4171/JEMS/1128