关于二项方程代数数逼近的Thue-Siegel定理的一个证明
A Proof of the Thue-Siegel Theorem about the Approximation of Algebraic Numbers for Binomial Equations
AI总结：
本文通过推广二项级数的连分数展开并构造代数逼近函数及相应积分，在不使用鸽巢原理的情况下，为二项方程代数数证明了带 Siegel 指数的 Thue 定理。
摘要：
1908年，Thue（1）证明了特殊形式的代数数 $ξ=\sqrt[n]{\frac{a}{b}}$ 对任意正数 $ε$，只能被有限多个有理数 $\frac{p}{q}$ 以如下不等式尖锐逼近：\\[ \left|ξ-\frac{p}{q}\right|\leq q^{-(\frac{n}{2}+1+ε)} .\\] 该证明使用了二项级数 $(1-z)^ω$ 的连分数展开，尽管这一用法可能在某种程度上是隐含的。在关于代数数逼近的后续工作（2,3）中，Thue 著名地改用了一种完全不同的工具，即 Dirichlet 抽屉原理，并进一步证明了上述结论对任意代数数都成立。Thue 的方法后来被 Siegel（4,5,6,7）推广，他证明了除其他结果外，对任意代数数，上述不等式中的指数 $\frac{n}{2}+1+ε$ 可以被替换为 $\frac{n}{m}+m-1+ε$，其中 $m$ 是某个自然数。本文展示了 Thue 在（1）中方法的一个推广；与 Thue 一样，我限制自己只考虑二项方程的根 $ξ=\sqrt[n]{\frac{a}{b}}$。二项级数的连分数展开被推广，并且用代数逼近函数代替了有理逼近函数。在此过程中，我完全按照我在指数函数方面的工作（8）进行。为逼近函数建立了积分；因此估计变得容易得多，并且可以毫无困难地、在不使用鸽巢原理的情况下，证明二项代数方程带 Siegel 指数的 Thue 定理。
英文摘要：
In 1908 Thue (1) showed that algebraic numbers of the special form $ξ=\sqrt[n]{\frac{a}{b}}$ can, for every positive $ε$, only be sharply approximated by finitely many rational numbers $\frac{p}{q}$ with the following inequality holding \[ \left|ξ-\frac{p}{q}\right|\leq q^{-(\frac{n}{2}+1+ε)} .\] The proof uses, if perhaps in a somewhat hidden way, the continued fraction expansion of the binomial series $(1-z)^ω$. 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 $\frac{n}{2}+1+ε$ could be replaced by $\frac{n}{m}+m-1+ε$, 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 $ξ=\sqrt[n]{\frac{a}{b}}$ 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.
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