Khintchine types of translated coordinate hyperplanes

TL;DR

This paper investigates the approximation properties of translated coordinate hyperplanes in Euclidean space, establishing a dichotomy similar to Khintchine's Theorem for these flat manifolds.

Contribution

It extends Khintchine-type results to flat manifolds, specifically translated coordinate hyperplanes, identifying conditions for measure-zero or full sets of rationally approximable points.

Findings

Identifies a Khintchine-type dichotomy for translated coordinate hyperplanes.

Characterizes when the set of rationally approximable points is null or full.

Provides conditions based on convergence or divergence of approximation series.

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.