Cayley Digraphs Associated to Arithmetic Groups

  • David CovertEmail author
  • Yesim Demiroğlu Karabulut
  • Jonathan Pakianathan
Original Paper


We explore a paradigm which ties together seemingly disparate areas in number theory, additive combinatorics, and geometric combinatorics including the classical Waring problem, the Furstenberg–Sárközy theorem on squares in sets of integers with positive density, and the study of triangles (also called 2-simplices) in finite fields. Among other results we show that if \({\mathbb {F}}_q\) is the finite field of odd order q, then every matrix in \(Mat_d({\mathbb {F}}_q), d \ge 2\) is the sum of a certain (finite) number of orthogonal matrices, this number depending only on d, the size of the matrix, and on whether q is congruent to 1 or 3 (mod 4), but independent of q otherwise.


Waring’s problem Cayley digraphs Orthogonal matrices General linear group Finite fields 

Mathematics Subject Classification

Primary 11P05 05C35 Secondary 15B10 



  1. 1.
    Bennett, M., Iosevich, A., Pakianathan, J.: Three-point configurations determined by subsets of \({\mathbb{F}}_q^2\) via the Elekes–Sharir paradigm. Combinatorica 34(6), 689–706 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bourgain, J., Katz, N., Tao, T.: A sum-product estimate in finite fields, and applications. GAFA 14, 27–57 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Chapman, J., Erdogan, M.B., Hart, D., Iosevich, A., Koh, D.: Pinned distance sets, k-simplices. Wolff’s exponent in finite fields and sum-product estimates. Math. Z. 271, 63–93 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Demiroğlu Karabulut, Y.: Waring’s Problem in Finite Rings (arXiv preprint). arXiv:1709.04428 (2017)
  5. 5.
    Demiroğlu Karabulut, Y.: Unit-graphs and special unit-digraphs on matrix rings. Forum Math. 30(6), 1397–1412 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Erdős, P.: Integral distances. Bull. Am. Math. Soc. 51, 248–250 (1946)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Furstenberg, H.: Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math. 71, 204–256 (1977)CrossRefGoogle Scholar
  8. 8.
    Hart, D., Iosevich, A.: Ubiquity of simplices in subsets of vector spaces over finite fields. Anal. Math. 34(1), 29–38 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hart, D., Iosevich, A., Solymosi, J.: Sum-product estimates in finite fields via Kloosterman Sums. Int. Math. Res. Not. IMRN 5, Art. ID rnm007 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Henriksen, M.: Two classes of rings generated by their units. J. Algebra 31(1), 182–193 (1974)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Iosevich, A., Rudnev, M.: Erdős distance problem in vector spaces over finite fields. Trans. Am. Math. Soc. 359, 6127–6142 (2007)CrossRefGoogle Scholar
  12. 12.
    Lang, S.: Algebra, 2nd edn. Addison-Wesley, Boston (1984)zbMATHGoogle Scholar
  13. 13.
    Sárközy, A.: On difference sets of sequences of integers III. Acta Math. Acad. Sci. Hungar. 31, 355–386 (1978)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Srivastava, A.: A survey of rings generated by units. Ann. Fac. Sci. Toulouse Math. (6) 19, 203–213 (2010) (Fascicule Special) Google Scholar
  15. 15.
    Tao, T.: The sum-product phenomenon in arbitrary rings. Contrib. Discrete Math. 4(2), 59–82 (2009)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Vaughan, R., Wooley, T.: Waring’s Problem: A Survey, Number Theory for the Millennium, III (Urbana, IL, 2000), pp. 301–340 (2002)Google Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  • David Covert
    • 1
    Email author
  • Yesim Demiroğlu Karabulut
    • 2
  • Jonathan Pakianathan
    • 3
  1. 1.University of Missouri - Saint LouisMissouriUSA
  2. 2.Harvey Mudd CollegeClaremontUSA
  3. 3.University of RochesterRochesterUSA

Personalised recommendations