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On a conjecture of Mahler

Published online by Cambridge University Press:  17 April 2009

V.C. Dumir  and
R.J. Hans-Gill
V.C. Dumir
Affiliation:
Centre for Advanced Study in Mathematics, Panjab University, Chandigarh, India.
R.J. Hans-Gill
Affiliation:
Centre for Advanced Study in Mathematics, Panjab University, Chandigarh, India.
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Abstract

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Let R be the field of real numbers. For α in R, let ‖α‖ be the distance of α from the nearest integer. The following conjecture of Kurt Mahler [Bull. Austral. Math. Soc. 14 (1976), 463–465] is proved.

Let m, n be two positive integers n ≥ 2m. Let S be a finite or infinite set of positive integers with the following properties:

(Q1) S contains the integers m, m+1, …, n−m;

(Q2) every element of S satisfies

Then

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 16 , Issue 1 , February 1977 , pp. 125 - 129
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]van der Corput, J.G., “Verallgemeinerung einer Mordellschen Beweismethode in der Geometrie der Zahlen”, Acta Arith. 1 (1936), 62–66.CrossRefGoogle Scholar
[2]Lekkerkerker, C.G., Geometry of numbers (Bibliotheca Mathematica, 8. Wolters-Noordhoff, Groningen; North-Holland, Amsterdam, London; 1969).Google Scholar
[3]Mahler, Kurt, “A theorem on diophantine approximations”, Bull. Austral. Math. Soc. 14 (1976), 463–465.CrossRefGoogle Scholar