Simultaneous rational approximations and related diophantine equations.

Abstract

We construct algebraic numbers for which especially sharp effective linear independence and simultaneous approximation measures can be proved. Similar results were obtained previously by Osgood and Fel'dman, but in certain special cases our results are more precise. We then apply our measures to diophantine equations. They give strong effective bounds for the size of solutions; for example in the case of two simultaneous Pell-type equations. We are able to give non-effective results for more general equations; for example in the case of Weierstrass elliptic curves with rational 2-torsion. Our methods involve the classical use of contour integration to construct a suitable sequence of approximating forms. However, this seems to be new in the context of the algebraic numbers studied by Osgood and Fel'dman.