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Cayley Digraphs Associated to Arithmetic Groups

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

Abstract

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.

Keywords

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

Mathematics Subject Classification

Primary 11P05 05C35 Secondary 15B10 

Notes

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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

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