Khintchine types of translated coordinate hyperplanes

Felipe A. Ramírez. "Khintchine types of translated coordinate hyperplanes." Acta Arithmetica 170.3 (2015): 243-273. <http://eudml.org/doc/278896>.

@article{FelipeA2015,
abstract = {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.},
author = {Felipe A. Ramírez},
journal = {Acta Arithmetica},
keywords = {Khintchine types; simultaneous approximation; three gaps theorem},
language = {eng},
number = {3},
pages = {243-273},
title = {Khintchine types of translated coordinate hyperplanes},
url = {http://eudml.org/doc/278896},
volume = {170},
year = {2015},
}

TY - JOUR
AU - Felipe A. Ramírez
TI - Khintchine types of translated coordinate hyperplanes
JO - Acta Arithmetica
PY - 2015
VL - 170
IS - 3
SP - 243
EP - 273
AB - 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.
LA - eng
KW - Khintchine types; simultaneous approximation; three gaps theorem
UR - http://eudml.org/doc/278896
ER -