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Explicit Lower Bounds for Rational Approximation to Algebraic Numbers
Author(s) -
Bennett Michael A.
Publication year - 1997
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611597000269
Subject(s) - mathematics , diophantine approximation , algebraic number , padé approximant , irrationality , mathematics subject classification , rational number , diophantine equation , algebra over a field , calculus (dental) , discrete mathematics , pure mathematics , mathematical analysis , rationality , medicine , dentistry , political science , law
In this paper, we apply Padé approximation methods to derive completely explicit measures of irrationality for certain classes of algebraic numbers. Our approach is similar to that taken previously by G.V. Chudnovsky but has some fundamental advantages with regards to determining implicit constants. Our general results may be applied to produce specific bounds of the flavour of | 2 3 − p q | > 1 4 ∼ q − 2.45and| 5 7 − p q | > 1 4 ∼ q − 4.43which we show to hold for any nonzero integers p and q . Further examples are tabulated and applications to Diophantine equations are briefly discussed as are other topics of related interest. 1991 Mathematics Subject Classification : primary 11J68, 11J82; secondary 11D41.