Mahler's classification of numbers compared with Koksma's
Yann Bugeaud · Acta Arithmetica · 2003
Mahler [8], in 1932, and Koksma [7], in 1939, introduced two related measures of the degree of approximation of a complex transcendental number ξ by algebraic numbers. Following Mahler [8], for any integer n ≥ 1, we denote by wn(ξ) the supremum of the exponents w for which $$ 0 < \left| {P\left( \xi \right)} \right| < H\left( P \right)^{ - \omega } $$ has infinitely many solutions in integer polynomials P(X) of degree at most n. Here, H(P) stands for the naive height of the polynomial P(X), that is, the maximum of the absolute values of its coefficients. Following Koksma [7], for any integer n ≥ 1, we denote by w n ξ the supremum of the exponents w for which $$ 0 < \left| {\xi - \alpha } \right| < H\left( \alpha \right)^{ - \omega - 1} $$ has infinitely many solutions in complex algebraic numbers α of degree at most n. Here, H(α) stands for the naive height of α, that is, the naive height of its minimal defining polynomial over Z. Clearly, the functions w1 and w 1 * coincide.