Vol. 92, No. 2, 1981

Recent Issues
Vol. 332: 1
Vol. 331: 1  2
Vol. 330: 1  2
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN 1945-5844 (electronic)
ISSN 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
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