An effective refinement of the exponent in Liouville's theorem

N. I. Fel'dman

Izvestiya. Mathematics , Tome 5 (1971) no. 5, pp. 985-1002

Résumé

For every algebraic number $\alpha$ of degree $n\geqslant3$ there exist effective positive constants $a$ and $C$ such that for any rational integers $q>0$ and $p$ we have $$ \biggl|\alpha-\frac pq\biggr|>Cq^{a-n}. $$ We also derive an effective boundary of the type $C_1m^{a_1}$ for the solutions of the Diophantine equation $f(x,y)=m$, where $f$ is a form of degree $\geqslant3$.