Effective approximation and Diophantine applications

Using the Thue–Siegel method, we obtain effective improvements on Liouville’s irrationality measure for certain one-parameter families of algebraic numbers, defined by equations of the type $(t - a)Q(t) + P(t) = 0$. We apply these to some corresponding Diophantine equations. We obtain bounds for the size of solutions, which depend polynomially on $|a|$, and bounds for the number of these solutions, which are independent of $a$ and in some cases even independent of the degree of the equation.