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Khintchine types of translated coordinate hyperplanes

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Ramírez F.

Acta Arithmetica

2015

tom 170

nr 3

243-273

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.

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1 Acta Arithmetica

1 Ramírez F.

1 2015

Wyniki wyszukiwania

Khintchine types of translated coordinate hyperplanes