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Distribution of residues in approximate subgroups of $ \mathbb{F}_p^*$

Authors: Norbert Hegyvári and François Hennecart
Journal: Proc. Amer. Math. Soc. 140 (2012), 1-6
MSC (2010): Primary 11B75
Published electronically: May 3, 2011
MathSciNet review: 2833512
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Abstract: We extend a result due to Bourgain on the uniform distribution of residues by proving that subsets of the type $ f(I)\cdot H$ are equidistributed (as $ p$ tends to infinity), where $ f$ is a polynomial, $ I$ is an interval of $ \mathbb{F}_p$ and $ H$ is an approximate subgroup of $ \mathbb{F}_p^*$ with size larger than polylogarithmic in $ p$.

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  • 1. Bourgain, J.; Mordell's exponential sum estimate revisited. J. Amer. Math. Soc. 18 (2005), no. 2, 477-499. MR 2137982 (2006b:11099)
  • 2. Bourgain, J.; On the distribution of the residues of small multiplicative subgroups of $ \mathbb{F}_p$. Israel J. of Math. 172 (2009), 61-74. MR 2534239 (2010i:11017)
  • 3. Bourgain, J.; Glibichuk, A.; Konyagin S.; Estimate for the number of sums and products and for exponential sums in fields of prime order. J. London Math. Soc. 73 (2006), 380-398. MR 2225493 (2007e:11092)
  • 4. Chang, M.-C.; Some problems in combinatorial number theory. Integers 8 (2008), no. 2, A1, 11 pp. MR 2438286 (2009d:11015)
  • 5. Dickson, L.E.; History of the theory of numbers. Chelsea Publishing, New York, 1971.
  • 6. Green, B.J; Ruzsa, I.Z.; Freiman's theorem in an arbitrary abelian group. J. London Math. Soc. (2) 75 (2007), 163-175. MR 2302736 (2007m:20087)
  • 7. Hegyvári, N.; Some remarks on multilinear exponential sums with an application. Preprint, 2010.
  • 8. Hegyvári, N.; Hennecart F.; Explicit construction of extractors and expanders, Acta Arith. 140 (2009), 233-249. MR 2564464 (2010j:11017)
  • 9. Sárközy, A.; On sums and products of residues modulo $ p$. Acta Arith. 118 (2005), 403-409. MR 2165553 (2006f:11023)
  • 10. Shkredov, I.D., On monochromatic solutions of some non linear equations. Preprint (2009).
  • 11. Shparlinski, I.E.; On the solvability of bilinear equations in finite fields. Glasg. Math. J. 50 (2008), no. 3, 523-529. MR 2451747 (2009j:11189)
  • 12. Tao, T; Vu, V.H.; Additive combinatorics. Cambridge Studies in Advanced Mathematics, 105. Cambridge University Press, Cambridge, 2006. MR 2289012 (2008a:11002)

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Additional Information

Norbert Hegyvári
Affiliation: Institute of Mathematics, Eötvös University, H-1117 Pázmány st. 1/c, Budapest, Hungary

François Hennecart
Affiliation: Université de Lyon and Université Jean-Monnet, 23, rue Michelon, 42023 Saint-Étienne, France

Received by editor(s): June 7, 2010
Received by editor(s) in revised form: October 26, 2010
Published electronically: May 3, 2011
Additional Notes: Research of the first author is partially supported by OTKA grants K 61908 and K 67676. He is grateful to the members of the LAMUSE (Laboratory of Mathematics of the University of Saint-Etienne) for their warm hospitality during his stay
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2011 American Mathematical Society

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