Reprint: On the approximation of algebraic numbers. I: On the greatest prime divisor of binary forms (1933)

Abstract

In 1908, Thue showed that, if $\zeta$ is a real algebraic number of degree $n$ and $\Theta$ is a positive number, the inequality $\left|\frac{p}{q}-\zeta \right|\le q^{-\left(\frac{n}{2}+1+\Theta \right)}$ has only finitely many rational solutions $p/q$. Siegel, in 1920, showed that one can replace the exponent $\frac{n}{2}+1+\Theta$ by $\beta ={\min }_{1\le s\le n-1}\left(\frac{n}{s+1}+s+\Theta \right).$

In this article, Mahler establishes the following $p$-adic extension of Siegel's result. Let $f(x)$ be an irreducible polynomial with rational integer coefficients of degree $n\ge 3$, let $P_1,P_2,\dots ,P_t$ be finitely many prime numbers and let $\zeta ,{\zeta }_1,{\zeta }_2,\dots ,{\zeta }_t$ be real zero of $f(x)$, a $P_1$-adic zero of $f(x)$, a $P_2$-adic zero of $f(x),\dots$, and a $P_t$-adic zero of $f(x)$, respectively. Mahler proves that, if $\beta$ is Siegel's exponent and $k\ge 1$ is fixed, the inequality \( \min\left{1,\left|\tfrac{p}{q}-\zeta\right|\right\} \prod_{\tau=1}^t \min{1,|p-q\zeta_\tau|_{P_\tau}\}\le k\, \max{|p|,|q|\}^{-\beta} \) has finitely many solutions in reduced rational numbers $p/q$.

Reprint of the author's paper [Math. Ann. 107, 691--730 (1933; Zbl 0006.10502; JFM 59.0220.01)]. For Part II see [Zbl 1465.11013].