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On the Distribution of the Sequence n2θ (mod 1)

Published online by Cambridge University Press:  20 November 2018

J. B. Friedlander  and
H. Iwaniec
J. B. Friedlander
Affiliation:
University of Toronto, Toronto, Ontario
H. Iwaniec
Affiliation:
Institute for Advanced Study, Princeton, New Jersey
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In a paper with the above title H. Heilbronn [4] proved that, given any real θ, there exist infinitely many positive integers n such that the distance ║θn2║ from θn2 to its nearest integral neighbour satisfies the bound

He actually proved a somewhat stronger statement which shows that the integers n occur with some regularity and he suggested that perhaps the exponent may be replaced by . This theorem has attracted considerable attention and spawned a number of generalizations (see [1], [7] and the references therein), yet no essential improvement has been given for the original problem (but see [6], [8]).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 2 , 01 April 1987 , pp. 338 - 344
Copyright
Copyright © Canadian Mathematical Society 1987

References

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