Mordell's exponential sum estimate revisited

Author:
J. Bourgain

Journal:
J. Amer. Math. Soc. **18** (2005), 477-499

MSC (2000):
Primary 11L07; Secondary 11T23

DOI:
https://doi.org/10.1090/S0894-0347-05-00476-5

Published electronically:
January 18, 2005

MathSciNet review:
2137982

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Abstract: The aim of this paper is to extend recent work of S. Konyagin and the author on Gauss sum estimates for large degree to the case of `sparse' polynomials. In this context we do obtain a nearly optimal result, improving on the works of Mordell and of Cochrane and Pinner. The result is optimal in terms of providing some power gain under conditions on the exponents in the polynomial that are best possible if we allow arbitrary coefficients. As in earlier work referred to above, our main combinatorial tool is a sum-product theorem. Here we need a version for product spaces for which the formulation is obviously not as simple as in the -case.

Again, the method applies more generally to provide nontrivial bounds on (possibly incomplete) exponential sums involving exponential functions. At the end of the paper, some applications of these are given to issues of uniform distribution for power generators in cryptography.

**[B]**J. Bourgain,*Estimates on exponential sums related to the Diffie-Hellman distributions*, to appear in GAFA.**[B-C]**J. Bourgain, M.-C. Chang,*Exponential sum estimates over subgroups and almost subgroups of**where**is composite with few prime factors*, submitted to Geom. Funct. Anal.**[B-G-K]**J. Bourgain, A. Glibichuk, S. Konyagin,*Estimate for the number of sums and products and for exponential sums in fields of prime order*, submitted to J. London Math. Soc.**[B-K]**Jean Bourgain and S. V. Konyagin,*Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order*, C. R. Math. Acad. Sci. Paris**337**(2003), no. 2, 75–80 (English, with English and French summaries). MR**1998834**, https://doi.org/10.1016/S1631-073X(03)00281-4**[B-K-T]**J. Bourgain, N. Katz, and T. Tao,*A sum-product estimate in finite fields, and applications*, Geom. Funct. Anal.**14**(2004), no. 1, 27–57. MR**2053599**, https://doi.org/10.1007/s00039-004-0451-1**[C-P1]**Todd Cochrane and Christopher Pinner,*An improved Mordell type bound for exponential sums*, Proc. Amer. Math. Soc.**133**(2005), no. 2, 313–320. MR**2093050**, https://doi.org/10.1090/S0002-9939-04-07726-3**[C-P2]**Todd Cochrane and Christopher Pinner,*Stepanov’s method applied to binomial exponential sums*, Q. J. Math.**54**(2003), no. 3, 243–255. MR**2013138**, https://doi.org/10.1093/qjmath/54.3.243**[F-P-S]**John B. Friedlander, Carl Pomerance, and Igor E. Shparlinski,*Period of the power generator and small values of Carmichael’s function*, Math. Comp.**70**(2001), no. 236, 1591–1605. MR**1836921**, https://doi.org/10.1090/S0025-5718-00-01282-5**[F-S]**John B. Friedlander and Igor E. Shparlinski,*On the distribution of the power generator*, Math. Comp.**70**(2001), no. 236, 1575–1589. MR**1836920**, https://doi.org/10.1090/S0025-5718-00-01283-7**[Go]**W. T. Gowers,*A new proof of Szemerédi’s theorem for arithmetic progressions of length four*, Geom. Funct. Anal.**8**(1998), no. 3, 529–551. MR**1631259**, https://doi.org/10.1007/s000390050065**[K-S]**Sergei V. Konyagin and Igor E. Shparlinski,*Character sums with exponential functions and their applications*, Cambridge Tracts in Mathematics, vol. 136, Cambridge University Press, Cambridge, 1999. MR**1725241****[Mor]**L.J. Mordell,*On a sum analogous to a Gauss sum*, Quart. J. Math. 3 (1932), 161-162.**[Na]**Melvyn B. Nathanson,*Additive number theory*, Graduate Texts in Mathematics, vol. 164, Springer-Verlag, New York, 1996. The classical bases. MR**1395371**

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Additional Information

**J. Bourgain**

Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540

DOI:
https://doi.org/10.1090/S0894-0347-05-00476-5

Received by editor(s):
July 16, 2004

Published electronically:
January 18, 2005

Article copyright:
© Copyright 2005
American Mathematical Society