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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
7.Vinogradov, I. M., The method of trigonometrical sums in the theory of numbers, (translated by Roth, K. F. and Davenport, Anne) (Interscience Publishers, 1954).Google Scholar