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.