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Khintchine types of translated coordinate hyperplanes

Volume 170 / 2015

Felipe A. Ramírez Acta Arithmetica 170 (2015), 243-273 MSC: Primary 11K60; Secondary 11K50. DOI: 10.4064/aa170-3-3


There has been great interest in developing a theory of “Khintchine types” for manifolds embedded in Euclidean space, and considerable progress has been made for curved manifolds. We treat the case of translates of coordinate hyperplanes, decidedly flat manifolds. In our main results, we fix the value of one coordinate in Euclidean space and describe the set of points in the fiber over that fixed coordinate that are rationally approximable at a given rate. We identify translated coordinate hyperplanes for which there is a dichotomy as in Khintchine's Theorem: the set of rationally approximable points is null or full, according to the convergence or divergence of the series associated to the desired rate of approximation.


  • Felipe A. RamírezDepartment of Mathematics
    University of York
    Heslington, York, UK
    Department of Mathematics and Computer Science
    Wesleyan University
    265 Church Street
    Middletown, CT 06459, U.S.A.

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