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On a variant of sum-product estimates and explicit exponential sum bounds in prime fields

Published online by Cambridge University Press:  01 January 2009

J. BOURGAIN  and
M. Z. GARAEV
J. BOURGAIN
Affiliation:
Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, U.S.A. e-mail: [email protected]
M. Z. GARAEV
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, Apartado Postal 61-3 (Xangari), C.P. 58089, Morelia, Michoacán, México. e-mail: [email protected]
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Abstract

Let Fp be the field of a prime order p and F*p be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the boundfor any subset A ⊂ Fp with 1 < |A| < p12/23. Then we apply our estimate to obtain explicit bounds for some exponential sums in Fp. We show that for any subsets X, Y, Z ⊂ F*p and any complex numbers αx, βy, γz with |αx| ≤ 1, |βy| ≤ 1, |γz| ≤ 1, the following bound holds:We apply this bound further to show that if H is a subgroup of F*p with |H| > p1/4, thenFinally we show that if g is a generator of F*p then for any M < p the number of solutions of the equationis less than . This implies that if p1/2 < M < p, then

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 1 , January 2009 , pp. 1 - 21
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

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