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Distribution of rational points on the real line

Published online by Cambridge University Press:  09 April 2009

T. K. Sheng
T. K. Sheng
Affiliation:
Faculty of Mathematics University of NewcastleNew South Wales, 2308., Australia
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It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 15 , Issue 2 , March 1973 , pp. 243 - 256
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Dickson, L. E., History of the Theory of Numbers Vol. 1 (Chelsea Publishing Company, 1952).Google Scholar
[2]Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers, (Oxford, 4th ed., 1960).Google Scholar
[3]Roth, K. F., ‘Rational approximations to algebraic numbers’, Mathematika 2 (1955), 1–20.CrossRefGoogle Scholar