Simultaneous rational approximation to binomial functions

Abstract

We apply Pade approximation techniques to deduce lower bounds for simultaneous rational approximation to one or more algebraic numbers. In particular, we strengthen work of Osgood, Fel′dman and Rickert, proving, for example, that max {∣∣√2− p1/q∣∣ , ∣∣√3− p2/q∣∣} > q−1.79155 for q > q0 (where the latter is an effective constant). Some of the Diophantine consequences of such bounds will be discussed, specifically in the direction of solving simultaneous Pell’s equations and norm form equations. 0. Introduction In 1964, Baker [1, 2] utilized the method of Pade approximation to hypergeometric functions to obtain explicit improvements upon Liouville’s theorem on rational approximation to algebraic numbers. By way of example, he showed that ∣∣∣∣ 3 2− pq ∣∣∣∣ > 10−6q−2.955 (0.1) for all positive integers p and q and used such bounds to solve related Diophantine equations. Chudnovsky [6] subsequently refined Baker’s results, primarily through a detailed analysis of the arithmetical properties of certain Pade approximants. Analogous to (0.1), he proved that ∣∣∣∣ 3 2− pq ∣∣∣∣ > q−2.42971 (0.2) for all integers p and q with q greater than some effectively computable constant q0. By working out the implicit constants in (0.2), Easton [8] deduced ∣∣∣∣ 3 2− pq ∣∣∣∣ > 6.6× 10−6q−2.795 for positive integers p and q (as well as related bounds for other cubic irrationalities). Similar results exist for simultaneous approximation to several algebraic numbers. In particular, Baker [3] derived bounds of the form max 1≤u≤m {∣∣∣∣θu − pu q ∣∣∣∣} > q−λ (0.3) Received by the editors June 30, 1994 and, in revised form, January 31, 1995. 1991 Mathematics Subject Classification. Primary 11J68, 11J82; Secondary 11D57.