Rational approximation to algebraic varieties and a new exponent of simultaneous approximation

TL;DR

This paper explores the limits of rational approximation on algebraic varieties, showing that well-approximable points are closely related to rational points, and introduces a new exponent for simultaneous approximation.

Contribution

It establishes a link between approximation quality and rational points on varieties and proposes a novel exponent for measuring simultaneous approximation.

Findings

Approximable points lie in the closure of rational points on the variety.

In cases with finitely many rational points, the closure equals the set of rational points.

Introduces a new exponent of simultaneous approximation.

Abstract

This paper deals with two main topics related to Diophantine approximation. Firstly, we show that if a point on an algebraic variety is approximable by rational vectors to a sufficiently large degree, the approximating vectors must lie in the topological closure of the rational points on the variety. In many interesting cases, in particular if the set of rational points on the variety is finite, this closure does not exceed the set of rational points on the variety itself. This result enables easier proofs of several known results as special cases. The proof can be generalized in some way and encourages to define a new exponent of simultaneous approximation. The second part of the paper is devoted to the study of this exponent.