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On the fractional parts of a set of points

Published online by Cambridge University Press:  26 February 2010

R. J. Cook
R. J. Cook
Affiliation:
University College, Cardiff
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Extract

Heilbronn [5] proved that for any ε > 0 there exists C(ε) such that for any real θ and N ≥ 1 there is an integer x satisfying

where ‖α‖ denotes the difference between α and the nearest integer, taken positively. The result is uniform in θ and so analogous to Dirichlet's inequality for the fractional parts of nθ. The result has been generalized to simultaneous approximations by Danicic [1] and Ming-chit Liu [6]. Here we shall extend the result to any finite number of simultaneous approximations when x2 is replaced by a k-th power.

Type
Research Article
Information
Mathematika , Volume 19 , Issue 1 , June 1972 , pp. 63 - 68
Copyright
Copyright © University College London 1972

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References

1.Danicic, I., “On the fractional parts of θx2 and φx2”, J. London Math. Soc., 34 (1959), 353–357.CrossRefGoogle Scholar
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5.Heilbronn, H., “On the distribution of the sequence n 2θ (mod 1)”, Quart. J. Math. Oxford, 19 (1948), 249–256.CrossRefGoogle Scholar
6.Liu, Ming-chit, “On the fractional parts of θnk and φn k”, Quart. J. Math. Oxford (2), 21 (1970), 481–486.CrossRefGoogle Scholar
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