Simultaneous rational approximations and related diophantine equations

Abstract

AbstractIn this paper we shall consider simultaneous approximations to algebraic numbers of the form for rational t1, …, tm, v. If t1, …, tm, are sufficiently close to 1 with respect to a common denominator we shall derive an effective inequality of the form max {|θ1 – p1 / q|, …, |θm – pm / q|} > cq–1–λ for all integers p1, …, pm, q with q > 0. The technique is used to derive the inequalityWe use this to show that the only solution to the simultaneous equationsare the trivial ones.