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Trigonometric sums of Heilbronn's type

Published online by Cambridge University Press:  24 October 2008

R. W. K. Odoni
R. W. K. Odoni
Affiliation:
Department of Mathematics, University of Exeter
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Extract

In problems of additive number theory one frequently needs to obtain a non-trivial estimate for the absolute value of a trigonometric sum of the type

where f(X) ε ℤ [X] and 1 ≤ m ε ℤ. The general procedure is first to reduce the estimation to the case where m is a prime power, by means of the Chinese Remainder Theorem. The case m = pr (p prime) can often be reduced to that of a lower power of p, by a substitution of the type x = u + vps (where 0 ≤ u < ps and 0 ≤ v < pr-s), followed by the use of a p-adic Taylor expansion f(u+psv) = f(u) +psvf′(u) +…. Frequently this gives T(f, pr) = 0 when r ≥ 2, or at least allows one to reduce to the case m = p. In the latter case an appeal to Weil's estimate

usually gives a good estimate for (0.1), at least if deg f = o(√p).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 98 , Issue 3 , November 1985 , pp. 389 - 396
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

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