Advertisement

The Ramanujan Journal

, Volume 46, Issue 2, pp 357–372 | Cite as

A note on the size of the set \(\varvec{A^2+A}\)

  • Norbert Hegyvári
  • François Hennecart
Article
  • 74 Downloads

Abstract

Let \(F(x,y,z)=xy+z\). We consider some properties of expansion of the polynomial F in different settings, namely in the integers and in prime fields. The main results concern the question of covering \(\{0,1,\ldots , N\}\) (resp. \(\mathbf {F}_p\)) by \(A^2+A\) with some thin sets A.

Keywords

Covering polynomial Expander Sum-product 

Mathematics Subject Classification

11B75 20G40 

Notes

Acknowledgements

We warmly thank the referee for pointing out various improvements in our results and their proofs.

References

  1. 1.
    Balog, A.: A note on sum-product estimates. Publ. Math. Debr. 79, 283–289 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bourgain, J.: More on the sum-product phenomenon in prime fields and its application. Int. J. Number Theory 1, 1–32 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chang, M.-C.: Sum and product of difference sets. Contrib. Discret. Math. 1, 47–56 (2006)MATHGoogle Scholar
  4. 4.
    Elekes, G.: On the number of sums and products. Acta Arith. 81, 365–367 (1997)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Elekes, G., Tóth, C.D.: Incidences of non too degenerate hyperplanes. In: Proceedings of the Twenty-First Annual Symposium on Computational Geometry (SCG’05), pp. 16–21. ACM, New York (2005)Google Scholar
  6. 6.
    Erdős, P.: Some remarks on number theory. Riveon Lematematika 9, 45–48 (1944)MathSciNetGoogle Scholar
  7. 7.
    Ford, K.: The distribution of integers with a divisor in a given interval. Ann. Math. 168, 367–433 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Halberstam, H., Roth, K.: Sequences, 2nd edn. Springer, New York (1983)CrossRefMATHGoogle Scholar
  9. 9.
    Hegyvári, N.: Symmetry sets, approximate groups. J. Comb. Number Theory 3, 27–32 (2011)MathSciNetMATHGoogle Scholar
  10. 10.
    Hegyvári, N., Hennecart, F.: Explicit construction of extractors and expanders. Acta Arith. 140, 233–249 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hegyvári, N., Hennecart, F.: A note on Freiman models in Heisenberg groups. Isr. J. Math. 189, 397–411 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Konyagin, S., Shkredov, I.D.: On sum sets of sets, having small product set. arXiv:1503.05771 (2015)
  13. 13.
    Murphy, B., Rudnev, M., Shkredov, I.D., Yacizi, E.A.: Growth estimates in positive characteristic via collisions. arXiv:1512.06613v1 (2015)
  14. 14.
    Nathanson, M.B., O’Bryant, K., Orosz, B., Ruzsa, I.Z., Silva, M.: Binary linear forms over finite sets of integers. Acta Arith. 129, 341–361 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Roche-Newton, O., Rudnev, M., Shkredov, I.D.: New sum-product type estimates over finite fields. arXiv:1408.0542 (2014)
  16. 16.
    Shkredov, I.D.: On monochromatic solutions of some non linear equations in \(({\mathbb{Z}}/p{\mathbb{Z}})^{*}\). Math. Notes 88, 603–611 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Shkredov, I.D.: On a question of A. Balog. Pac. J. Math. 280, 227–240 (2016)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Singer, J.: A theorem in finite projective geometry and some applications to number theory. Trans. Am. Math. Soc. 43, 377–385 (1938)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Vinh, L.A.: Szemeredi-Trotter type theorem and sum-product estimate in finite fields. Eur. J. Comb. 32, 1177–1181 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.ELTE TTK, Institute of MathematicsEötvös UniversityBudapestHungary
  2. 2.Univ Lyon, UJM-Saint-Étienne, CNRS, ICJ UMR 5208Saint-ÉtienneFrance

Personalised recommendations