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On the fractional parts of a set of points
Published online by Cambridge University Press: 26 February 2010
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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.
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- Copyright © University College London 1972
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