On algebraic approximations of certain algebraic numbers. (Q1403931)

The main result of the present paper consists of an effective improvement on Liouville's bound for the approximations by rational numbers to algebraic numbers of a special type. Namely, the author considers \(n\)-th roots of imaginary quadratic irrationals. As a corollary of his estimates, the author obtains a bound for the primitive solutions of the Thue-inequality \[ | BX^4-AX^3Y-6BX^2Y^2+AXY^3+BY^4| \leq N. \] Whenever \(A>308 B^4\) every primitive solution \((x,y)\in {\mathbb Z}^2\) to the above inequality verifies \[ x^2+y^2\leq\max\left\{{25 A^2\over 64 B^2},{4N^2\over A^2}\right\}. \] This is a generalization of previous results in \textit{G. Lettl}, \textit{A. Pethő} and \textit{P. Voutier} [Trans. Am. Math. Soc. 351, 1871--1894 (1999; Zbl 0920.11041)].