A Proof of the Thue-Siegel Theorem about the Approximation of Algebraic Numbers for Binomial Equations

TL;DR

This paper generalizes Thue's methods for approximating algebraic numbers of binomial form, providing a new proof of the Thue-Siegel theorem using integral-based approximation functions, simplifying the process and avoiding the pigeonhole principle.

Contribution

It introduces a generalized continued fraction expansion for binomial series and uses algebraic approximation functions, extending Thue's approach to prove the theorem more straightforwardly.

Findings

Thue's theorem is proved using integral estimates for approximation functions.

The method avoids the pigeonhole principle, simplifying the proof.

The approach generalizes to algebraic numbers of binomial form.

Abstract

In 1908 Thue (1) showed that algebraic numbers of the special form $\xi =\sqrt[n]{\frac{a}{b}}$ can, for every positive $\epsilon$, only be sharply approximated by finitely many rational numbers $\frac{p}{q}$ with the following inequality holding \[ \left|\xi -\frac{p}{q}\right|\leq q^{-(\frac{n}{2}+1+\epsilon)} .\] The proof uses, if perhaps in a somewhat hidden way, the continued fraction expansion of the binomial series $(1-z)^\omega$. In further work about the approximation of algebraic numbers (2,3) famously Thue used instead a completely different tool, the drawer method of Dirichlet, and showed further that the above statement holds for any algebraic number. Thue's methods were later generalized by Siegel(4,5,6,7) who showed, among other things, that for every algebraic number in the above inequality the exponent $\frac{n}{2}+1+\epsilon$ could be replaced by $\frac{n}{m}+m-1+\epsilon$, where $m$ is some natural number. This note demonstrates a generalization of Thue's methods in (1); Like Thue I restrict myself to the roots $\xi =\sqrt[n]{\frac{a}{b}}$ of the binomial equations. The continued fraction expansion of the binomial series is generalized and algebraic approximation functions are given instead of rational approximation functions. In doing so I proceed exactly as in my work on the exponential function(8). Integrals are set up for the approximation functions; thus the estimates become much easier and you can prove Thue's theorem with Siegel's Exponents for the binomial algebraic equations without difficulty and without use of the pigeonhole principle.