Density and ramsey type results on algebraic equations with restricted solution sets

Abstract

In earlier papers Sárközy studied the solvability of the equations

$$a + b = cd, a \in \mathcal{A}, b \in \mathcal{B}, c \in \mathcal{C}, d \in \mathcal{D},$$

resp.

$$ab + 1 = cd, a \in \mathcal{A}, b \in \mathcal{B}, c \in \mathcal{C}, d \in \mathcal{D}$$

where \(\mathcal{A},\mathcal{B},\mathcal{C},\mathcal{D}\) are “large” subsets of \(\mathbb{F}_p\). Later Gyarmati and Sárközy generalized and extended these problems by studying these equations and also other algebraic equations with restricted solution sets over finite fields. Here we will continue the work by studying further special equations over finite fields and also algebraic equations with restricted solution sets over the set of the integers, resp. rationals. We will focus on the most interesting cases of algebraic equations with 3, resp. 4 variables. In the cases when there are no “density results” of the above type, we will be also looking for Ramsey type results, i.e., for monochromatic solutions of the given equation. While in the earlier papers character sum estimates were used, now combinatorial tools dominate.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    F. Behrend: On sets of integers which contain no three terms in arithmetical progression, Proc. Nat. Acad. Sci. U. S. A. 32 (1946), 331–332.

    MathSciNet  MATH  Article  Google Scholar 

  2. [2]

    J. Bourgain: On triples in arithmetic progressions, Geom. Funct. Anal. 9 (1999), 968–984.

    MathSciNet  MATH  Article  Google Scholar 

  3. [3]

    P. Erdös and A. Sárközy: On a conjecture of Roth and some related problems, II, in: Number Theory, Proceedings of the First Conference of the Canadian Number Theory Association (held at Banff Center, Banff, Alberta, April 17–27, 1988), ed. R. A. Mollin, Walter de Gruyter, Berlin-New York, 1990; 125–138.

    Google Scholar 

  4. [4]

    P. Erdös, A. Sárközy and V. T. Sós: On a conjecture of Roth and some related problems, I, in: Irregularities of Partitions, eds. G. Halász and V. T. Sós, Algorithms and Combinatorics 8, Springer-Verlag, Berlin-Heidelberg-New York, 1989; 47–59.

    Google Scholar 

  5. [5]

    K. Gyarmati: On a problem of Diophantus, Acta Arith. 97 (2001), 53–65.

    MathSciNet  MATH  Article  Google Scholar 

  6. [6]

    K. Gyarmati and A. Sárközy: Equations in finite fields with restricted solution sets, I (Character sums), Acta Math. Hungar. 118 (2008), 129–148.

    MathSciNet  MATH  Article  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]

    N. Hindman: Monochromatic sums equal to products in N, Integers 11A (2011), Article 10, 10.

  9. [9]

    K. F. Roth: On certain sets of integers, J. London Math. Soc. 28 (1953), 104–109.

    MathSciNet  MATH  Article  Google Scholar 

  10. [10]

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

    MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    A. Sárközy: On products and shifted products of residues modulo p, Integers 8 (2008), Article 9, 8.

  12. [12]

    J. Schur: Über die Kongruenz x m+y mz m (mod p), Jahresber. Deutschen Math. Verein. 25 (1916), 114–117.

    MATH  Google Scholar 

  13. [13]

    B. L. van der Waerden: Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15 (1927), 212–216.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Péter Csikvári.

Additional information

Research partially supported by the Hungarian National Foundation for Scientific Research, Grants No. T 043623, T 043631, T 049693 and PD72264 and the János Bolyai Research Fellowship.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Csikvári, P., Gyarmati, K. & Sárközy, A. Density and ramsey type results on algebraic equations with restricted solution sets. Combinatorica 32, 425–449 (2012). https://doi.org/10.1007/s00493-012-2697-9

Download citation

Mathematics Subject Classification (2000)

  • 11B75