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A linear set of infinite measure with no two points having integral ratio

Published online by Cambridge University Press:  26 February 2010

J. A. Haight
J. A. Haight
Affiliation:
Department of Mathematics, Westfield College, London, N.W.3.
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Extract

It is not difficult to construct an unbounded set E on the positive real line such that, if x1, x2 belong to E, then x1/x2 is never equal to an integer. Our object is to show that it is possible to find such a set E which is measurable and of infinite Lebesgue measure. We were led to consider this problem through a study of those sets E, which are of infinite measure, yet, for each x > 0, nx є E for only a finite number of integers n. Sets of this type were first discovered by C. G. Lekkerkerker [2]. The set that we consider has both these properties. For another result on lattice points in sets of infinite measure see [1].

Type
Research Article
Information
Mathematika , Volume 17 , Issue 1 , June 1970 , pp. 133 - 138
Copyright
Copyright © University College London 1970

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References

1.Davenport, H. and Erdős, P., “A theorem on uniform distribution”, Publ. Math. Inst. Hung. Acad., A, 8 (1963), 3–11.Google Scholar
2.Lekkerkerker, C. G., “Lattice points in unbounded point sets”, I. Indag. Math., 20 (1958), 197–205.CrossRefGoogle Scholar