Vol. 92, No. 2, 1981

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Approximations to real algebraic numbers by algebraic numbers of smaller degree

John F. Morrison

Vol. 92 (1981), No. 2, 403–413
Abstract

If β is an algebraic number of degree n + 1 then the number of solutions α, with α algebraic of degree at most n, to the inequalities

|β − α| < φ (H (α)), 1 ≦ H(α) ≦ B
(1)

is studied using methods developed by Schmidt and Adams for counting solutions to inequalities involving linear forms. In (1) H(α) is a height function which differes slightly from the usual height and φ is a function which decreases to zero.

If φ(y)yn+1 →∞ as y →∞ then the number of solutions is given as an integral plus an error term. If φ(y)yn+1 is constant then the number of solutions is either bounded or asymptotic to C log B for some constant C.

Mathematical Subject Classification
Primary: 10F25, 10F25
Milestones
Received: 9 November 1979
Published: 1 February 1981
Authors
John F. Morrison