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On a class of diophantine inequalities

Published online by Cambridge University Press:  17 April 2009

Kurt Mahler
Kurt Mahler
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
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Abstract

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As a special case of more general results, it is proved in this note that, if α is any real number and δ any positive number, then there exists a positive integer X such that the inequality

has infinitely many solutions in positive integers h and Yh.

The method depends on the study of infinite sequences of real linear forms in a fixed number of variables. It has relations to that used by Kronecker in the proof of his classical theorem and can be generalised.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 8 , Issue 2 , April 1973 , pp. 247 - 259
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Pisot, Charles, “La repartition modulo 1 et les nombres algébriques”, Ann. Souola Norm. Sup. Pisa, Sci. Fis. Math. (2) 7 (1938), 205–248.Google Scholar
[2]Salem, Raphael, Algebraic numbers and Fourier analysis (D.C. Heath and Co., Boston, Massachusetts, 1963).Google Scholar