Effective Irrationality Measures for Certain Algebraic Numbers

A result of Chudnovsky concerning rational approximation to certain algebraic numbers is reworked to provide a quantitative result in which all constants are explicitly given. More particularly, Padé approximants to the function (1 jc)1/3 are employed to show, for certain integers a and b, that |(a/fc) p/q\> cq~* when q > 0. Here, c and k are given as functions of a and b only. In 1964 Baker [1], improving a technique used by Siegel [8], was able to obtain effective irrationality measures for the function (1 x)m/" evaluated at certain rational points. In particular, he was able to show that for integers p, q we have (1) \21/i-p/q\> lfrV2-955 whentf>0. The technique was further refined by Chudnovsky [2] whose results, when applied to 21/3, imply that for any e > 0 there exists a positive integer q0(e) such that for integers p, q we have (2) \lW -p/q\> q-Q*»+* when q>q0(e). Chudnovsky's result is effective in the sense that it is possible in principle to work through the proof and compute, for any particular value of e, a <70(e) for which (2) holds. However, Chudnovsky does not undertake such computations. In this article we rework Baker's proof using Chudnovsky's refinement, together with a Chebyshev-type result for primes in arithmetical progressions due to McCurley [6], and obtain the following quantitative result: Theorem. Let a, b be integers with 0 < b < a. Define d by (0 ifH(a-b), (3) d=ll if3\\(a-b), \3/2 otherwise. Further, define \, k, c and q0 by (4) A = (.2328)3V/2-¿>1/2)"2, Received May 9, 1985. 1980 Mathematics Subject Classification. Primary 10F25. 613 ©1986 American Mathematical Society 0025-5718/86 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use