Combinatorial problems in finite fields and Sidon sets

Abstract

We use Sidon sets to present an elementary method to study some combinatorial problems in finite fields, such as sum product estimates, solvability of some equations and the distribution of their solutions. We obtain classic and more recent results avoiding the use of exponential sums, the usual tool to deal with these problems.

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References

  1. [1]

    P. Csikvári, K. Gyarmati and A. Sárközy: Density and Ramsey type results on algebraic equations with restricted solution sets, Combinatorica 32 (2012), 425–449.

    MathSciNet  Article  Google Scholar 

  2. [2]

    M. Z. Garaev: The sum-product estimate for large subsets of prime fields. Proc. Amer. Math. Soc. 136 (2008), 2735–2739.

    MathSciNet  MATH  Article  Google Scholar 

  3. [3]

    M. Z. Garaev: On the logarithmic factor in error term estimates in certain additive congruence problems, Acta Arithmetica 124 (2006), 33–40.

    MathSciNet  Article  Google Scholar 

  4. [4]

    M. Z. Garaev and Ka-Lam Kueh: Distribution of special sequences modulo a large prime, International Journal of Mathematics and Mathematical Sciences 50 (2003), 3189–3194.

    MathSciNet  Article  Google Scholar 

  5. [5]

    M. Z. Garaev and C. Shen: On the size of the set A(A + 1), Mathematische Zeitschrift 265 (2010), 125–132.

    MathSciNet  MATH  Article  Google Scholar 

  6. [6]

    V. C. García: A note on an additive problem with powers of a primitive root, Bol. Soc. Mat. Mexicana (3) 11 (2005), 1–4.

    MathSciNet  MATH  Google Scholar 

  7. [7]

    K. Gyarmati and A. Sárközy: Equations in finite fields with restricted solution sets, II. (Algebraic equations.) Acta Math. Hungar. 119 (2008), 259–280.

    MathSciNet  MATH  Article  Google Scholar 

  8. [8]

    D. Hart, L. Li, Ch-Y. Shen: Fourier analysis and expanding phenomena in finite fields, Proc. Amer. Math. Soc. 32 (2011), 1177–1181.

    Google Scholar 

  9. [9]

    S. V. Konyagin: Bounds of exponential sums over subgroups and Gauss sums, Proc 4th Intern. Conf. Modern Problems of Number Theory and Its Applications, Moscow Lomonosov State Univ., Moscow, (2002) 86–114 (in Russian).

    Google Scholar 

  10. [10]

    Z. Rudnik and A. Zaharescu: The distribution of spacing between small powers of a primitive root, Israel Journal of Mathematics 120 (2000), 271–287.

    MathSciNet  Google Scholar 

  11. [11]

    A. Sárközy: On sums and products on residues modulo p, Acta Arithmetica 118 (2005), 403–409.

    MathSciNet  MATH  Article  Google Scholar 

  12. [12]

    A. Sárközy: On products and shifted products of residues modulo p, Proceedings of CANT 2005, Integers 8 (2008).

  13. [13]

    I. D. Shkredov: On monochromatic solutions of some nonlinear equations in ℤ/ℤp, Mathematical Notes 88 (2010), 603–611.

    MathSciNet  MATH  Article  Google Scholar 

  14. [14]

    J. Solymosi: Incidences and the spectra of graphs, Combinatorial Number Theory and Additive Group Theory, Advanced Courses in Mathematics — CRM Barcelona Birkhäuser Basel (2009), 299–314.

    Google Scholar 

  15. [15]

    L. A. Vinh: Szemeredi-Trotter type theorem and sum-product estimate in finite fields, to appear in Europ. J. of Comb.

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Correspondence to Javier Cilleruelo.

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Cilleruelo, J. Combinatorial problems in finite fields and Sidon sets. Combinatorica 32, 497–511 (2012). https://doi.org/10.1007/s00493-012-2819-4

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Mathematics Subject Classification (2010)

  • 11B83