  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.
