Thue's fundamental theorem. II: Further refinements and examples (Q897537)

In this paper, the author sharpens and simplifies earlier results he obtained in Part I [Acta Arith. 143, No. 2, 101--144 (2010; Zbl 1292.11083)] and deduces explicit irrationality measures for certain roots of polynomials of the form \((X-\sqrt{t})^n+(X+\sqrt{t})^n\) where \(n\geq 4\) and \(t\) is a negative integer. Let \(t\) be a rational integer, not a square. Denote by \(\sigma\) the non trivial element of \({\mathrm {Gal}}({\mathbb Q}(\sqrt{t})/{\mathbb Q})\). Let \(\beta\) and \(\gamma\) be algebraic integers in \({\mathbb Q}(\sqrt{t})\), \(u_1\), \(u_2\) rational integers, \(m\), \(n\) relatively prime positive rational integers with \(0<m<n\). Define \[ \delta=\left(\frac{\eta}{\sigma(\eta)}\right)^{m/n} \quad\text{and}\quad \alpha=\frac{ \beta\delta\pm\sigma(\beta)} { \gamma\delta\pm\sigma(\gamma)} \cdotp \] Under suitable general assumptions, the author gives a very sharp irrationality measure for \(\alpha\). In the same vein, he gives a similar estimate for numbers \((a/b)^{m/n}\) when \(a\), \(b\) are algebraic integers in a quadratic field. Among several remarkable explicit examples given in the text, we quote only the following: \[ \left| \sqrt{77}\tan\left(\frac{2\pi}{7}\right)-\frac{p}{q}\right| >\frac{3}{1000} q^{-3.49}. \]