# Effective approximation and Diophantine applications

Dill, Gabriel A.. (2017) Effective approximation and Diophantine applications. Acta Arithmetica, 177. pp. 169-199.

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Official URL: http://edoc.unibas.ch/53777/

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.