Effective approximation and Diophantine applications
Volume 177 / 2017
Acta Arithmetica 177 (2017), 169-199 MSC: Primary 11D41; Secondary 11D45, 11D57, 11J68. DOI: 10.4064/aa8430-9-2016 Published online: 22 December 2016
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.