Enrico Bombieri proved that the ABC Conjecture implies Roth's theorem in 1994. This paper concerns the other direction. In making use of Bombieri's and Van der Poorten's explicit formula for the coefficients of the regular continued fractions of algebraic numbers, we prove that Roth's theorem...

This paper argues that the distribution of the coefficients of the regular continued fraction should be considered for each algebraic number of degree 2 separately. For random numbers the coefficients are distributed by the Gauss-Kuzmin distribution (also called Khinchin's law). We apply the...