# Analytic Number Theory: The Number of Algebraic Numbers of Given Degree Approximating a Given Algebraic Number

```@inproceedings{Evertse1997AnalyticNT,
title={Analytic Number Theory: The Number of Algebraic Numbers of Given Degree Approximating a Given Algebraic Number},
author={Jan-Hendrik Evertse},
year={1997}
}```
has only finitely many solutions. Roth’s proof is by contradiction. Assuming that (1.1) has infinitely many solutions, Roth constructed an auxiliary polynomial in a large number of variables, k say, of which all low order partial derivatives vanish in a point (x1/y1, . . . , xk/yk) for certain solutions (x1, y1), . . . , (xk, yk) of (1.1), and then showed, using a non-vanishing result now known as Roth’s lemma, that this is not possible.
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