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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
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
Copyright
Copyright © University College London 1972

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References

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