The sumproduct estimate for large subsets of prime fields
Author:
M. Z. Garaev
Journal:
Proc. Amer. Math. Soc. 136 (2008), 27352739
MSC (2000):
Primary 11B75, 11T23
Published electronically:
April 14, 2008
MathSciNet review:
2399035
Fulltext PDF Free Access
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Abstract: Let be the field of prime order It is known that for any integer one can construct a subset with such that One of the results of the present paper implies that if with then
 1.
J. Bourgain, The sumproduct theorem in with arbitrary, preprint.
 2.
J.
Bourgain, More on the sumproduct phenomenon in prime fields and
its applications, Int. J. Number Theory 1 (2005),
no. 1, 1–32. MR 2172328
(2006g:11041), http://dx.doi.org/10.1142/S1793042105000108
 3.
J.
Bourgain and M.C.
Chang, Exponential sum estimates over subgroups and almost
subgroups of ℤ_{ℚ}*, where ℚ is composite with few
prime factors, Geom. Funct. Anal. 16 (2006),
no. 2, 327–366. MR 2231466
(2007d:11093), http://dx.doi.org/10.1007/s0003900605587
 4.
J.
Bourgain, A.
A. Glibichuk, and S.
V. Konyagin, Estimates for the number of sums and products and for
exponential sums in fields of prime order, J. London Math. Soc. (2)
73 (2006), no. 2, 380–398. MR 2225493
(2007e:11092), http://dx.doi.org/10.1112/S0024610706022721
 5.
J.
Bourgain, N.
Katz, and T.
Tao, A sumproduct estimate in finite fields, and
applications, Geom. Funct. Anal. 14 (2004),
no. 1, 27–57. MR 2053599
(2005d:11028), http://dx.doi.org/10.1007/s0003900404511
 6.
M.C. Chang, Some problems in combinatorial number theory, preprint.
 7.
P.
Erdős and E.
Szemerédi, On sums and products of integers, Studies in
pure mathematics, Birkhäuser, Basel, 1983, pp. 213–218. MR 820223
(86m:11011)
 8.
M.
Z. Garaev, An explicit sumproduct estimate in
𝔽_{𝕡}, Int. Math. Res. Not. IMRN 11
(2007), Art. ID rnm035, 11. MR 2344270
(2008g:11038)
 9.
Derrick
Hart, Alex
Iosevich, and Jozsef
Solymosi, Sumproduct estimates in finite fields via Kloosterman
sums, Int. Math. Res. Not. IMRN 5 (2007), Art. ID
rnm007, 14. MR
2341599 (2008i:11037), http://dx.doi.org/10.1093/imrn/rnm007
 10.
N. H. Katz and Ch.Y. Shen, A slight improvement to Garaev's sum product estimate, preprint.
 11.
N. H. Katz and Ch.Y. Shen, Garaev's inequality in fields not of prime order, preprint.
 12.
József
Solymosi, On the number of sums and products, Bull. London
Math. Soc. 37 (2005), no. 4, 491–494. MR 2143727
(2006c:11021), http://dx.doi.org/10.1112/S0024609305004261
 13.
I.
M. Vinogradov, An introduction to the theory of numbers,
Pergamon Press, London & New York, 1955. Translated by H. Popova. MR 0070644
(17,13a)
 14.
V. Vu, Sumproduct estimates via directed expanders, arXiv:0705.0715v1 [math.CO].
 1.
 J. Bourgain, The sumproduct theorem in with arbitrary, preprint.
 2.
 J. Bourgain, More on the sumproduct phenomenon in prime fields and its applications, Int. J. Number Theory 1 (2005), 132. MR 2172328 (2006g:11041)
 3.
 J. Bourgain and M.C. Chang, Exponential sum estimates over subgroups and almost subgroups of , where is composite with few prime factors, Geom. Funct. Anal. 16 (2006), 327366. MR 2231466 (2007d:11093)
 4.
 J. Bourgain, A. A. Glibichuk and S. V. Konyagin, Estimates for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. (2) 73 (2006), 380398. MR 2225493 (2007e:11092)
 5.
 J. Bourgain, N. Katz and T. Tao, A sumproduct estimate in finite fields, and applications, Geom. Funct. Anal. 14 (2004), 2757. MR 2053599 (2005d:11028)
 6.
 M.C. Chang, Some problems in combinatorial number theory, preprint.
 7.
 P. Erdös and E. Szemerédi, On sums and products of integers. Studies in pure mathematics, 213218, Birkhäuser, Basel, 1983. MR 820223 (86m:11011)
 8.
 M. Z. Garaev, An explicit sumproduct estimate in , Int. Math. Res. Notices (2007), no. 11, Art. ID rnm035. MR 2344270
 9.
 D. Hart, A. Iosevich and J. Solymosi, Sumproduct estimates in finite fields via Kloosterman sums, Int. Math. Res. Notices (2007), no. 5, Art. ID rnm007. MR 2341599
 10.
 N. H. Katz and Ch.Y. Shen, A slight improvement to Garaev's sum product estimate, preprint.
 11.
 N. H. Katz and Ch.Y. Shen, Garaev's inequality in fields not of prime order, preprint.
 12.
 J. Solymosi, On the number of sums and products, Bull. London Math. Soc. 37 (2005), 491494. MR 2143727 (2006c:11021)
 13.
 I. M. Vinogradov, An introduction to the theory of numbers, Pergamon Press, London and New York, 1955. MR 0070644 (17:13a)
 14.
 V. Vu, Sumproduct estimates via directed expanders, arXiv:0705.0715v1 [math.CO].
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Additional Information
M. Z. Garaev
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, Apartado Postal 613 (Xangari), C.P. 58089, Morelia, Michoacán, México
Email:
garaev@matmor.unam.mx
DOI:
http://dx.doi.org/10.1090/S0002993908093866
PII:
S 00029939(08)093866
Keywords:
Sumproduct estimates,
prime field,
number of solutions.
Received by editor(s):
June 26, 2007
Published electronically:
April 14, 2008
Communicated by:
Ken Ono
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
