Khintchine types of translated coordinate hyperplanes
Seminar Room 2, Newton Institute Gatehouse
AbstractThis talk is about the problem of simultaneously approximating a tuple of real numbers by rationals, when one of the real numbers has been prescribed. This corresponds to describing the set of rationally approximable points on translates of coordinate hyperplanes in Euclidean space. It is expected that under a divergence condition on the desired rate of approximation, we should be able to assert that almost every point on the hyperplane is rationally approximable at that rate, like in Khintchine's Theorem. We will discuss some positive results in this direction. These can be seen as living in the degenerate counterpart to work of Beresnevich and Beresnevich--Dickinson--Velani, where similar results were achieved for non-degenerate submanifolds of Euclidean space.
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