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On the fractional parts of the sum of powers of rational numbers

Published online by Cambridge University Press:  26 February 2010

Hans Peter Schlickewei
Hans Peter Schlickewei
Affiliation:
Mathematisches Institut der Universität, 7800 Freiburg i. Breisgau, Hebelstr. 29, Germany
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Extract

We shall prove the following

THEOREM 1. Let α1, …, αn be any positive algebraic numbers and let u1…, un, ν be positive integers, relatively prime in pairs, such that ν ≥ 2 and ui > v for at least one i (1 ≤ i ≤ n). Then for every ε > 0 there are only a finite number of positive integers v such that the inequality

is satisfied, where for real α we understand by ‖α‖ the distance of α from the nearest integer.

Type
Research Article
Information
Mathematika , Volume 22 , Issue 2 , December 1975 , pp. 154 - 155
Copyright
Copyright © University College London 1975

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References

1.Mahler, K.. “On the fractional parts of the powers of a rational number (II)”, Mathematika, 4 (1957), 122–124.CrossRefGoogle Scholar
2.Ridout, D.. “Rational approximations to algebraic numbers”, Mathematika, 4 (1957), 125–131.CrossRefGoogle Scholar
3.Schlickewei, H. P.. “On products of special linear forms with algebraic coefficients”, to appear.Google Scholar