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The values of a trigonometrical polynomial at well spaced points

Published online by Cambridge University Press:  26 February 2010

H. Davenport  and
H. Halberstam
H. Davenport
Affiliation:
University of Michigan.
H. Halberstam
Affiliation:
Universities of Cambridge and Nottingham.
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Extract

1. A study of the recent papers of Roth and Bombieri on the large sieve has led us to the following simple result on the sum of the squares of the absolute values of a trigonometric polynomial at a finite set of points.

Type
Research Article
Information
Mathematika , Volume 13 , Issue 1 , June 1966 , pp. 91 - 96
Copyright
Copyright © University College London 1966

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References

† Mathematika, 12 (1965), 1–9.CrossRefGoogle Scholar

‡ Mathematika, 12 (1965), 201–225.CrossRefGoogle Scholar

§ We denote by ‖θ‖ the distance from θ to the nearest integer.

If δN is sufficiently small, the factor on the right of (3) can be improved to (1 +ε) δ-1 by replacing ψ(x) in (8) by the function which is 1 for ‖x‖<η and 0 otherwise. We are indebted to Dr. H. Stark for the remark that the constant 2·2 cannot be replaced by a number less than 2–8π-2. This is shown by the following example (taking N to be large):

page92 note05 † Functions of this type played an important part in Roth's argument.

page96 note06 † Zygmund, Trigonometric Series (2nd ed.), II, 101.