Khintchine types of translated coordinate hyperplanes

Felipe A. Ramírez¹

¹ Department of Mathematics University of York Heslington, York, UK and Department of Mathematics and Computer Science Wesleyan University 265 Church Street Middletown, CT 06459, U.S.A.

Acta Arithmetica, Tome 170 (2015) no. 3, pp. 243-273.

Résumé

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.

Détail

DOI : 10.4064/aa170-3-3

Keywords: there has great interest developing theory khintchine types manifolds embedded euclidean space considerable progress has made curved manifolds treat translates coordinate hyperplanes decidedly flat manifolds main results fix value coordinate euclidean space describe set points fiber fixed coordinate rationally approximable given rate identify translated coordinate hyperplanes which there dichotomy khintchines theorem set rationally approximable points null full according convergence divergence series associated desired rate approximation