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.
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References
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)
Bourgain, J., Katz, N., Tao, T.: A sum-product estimate in finite fields, and applications. GAFA 14, 27–57 (2004)
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)
Demiroğlu Karabulut, Y.: Waring’s Problem in Finite Rings (arXiv preprint). arXiv:1709.04428 (2017)
Demiroğlu Karabulut, Y.: Unit-graphs and special unit-digraphs on matrix rings. Forum Math. 30(6), 1397–1412 (2018)
Erdős, P.: Integral distances. Bull. Am. Math. Soc. 51, 248–250 (1946)
Furstenberg, H.: Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math. 71, 204–256 (1977)
Hart, D., Iosevich, A.: Ubiquity of simplices in subsets of vector spaces over finite fields. Anal. Math. 34(1), 29–38 (2008)
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)
Henriksen, M.: Two classes of rings generated by their units. J. Algebra 31(1), 182–193 (1974)
Iosevich, A., Rudnev, M.: Erdős distance problem in vector spaces over finite fields. Trans. Am. Math. Soc. 359, 6127–6142 (2007)
Lang, S.: Algebra, 2nd edn. Addison-Wesley, Boston (1984)
Sárközy, A.: On difference sets of sequences of integers III. Acta Math. Acad. Sci. Hungar. 31, 355–386 (1978)
Srivastava, A.: A survey of rings generated by units. Ann. Fac. Sci. Toulouse Math. (6) 19, 203–213 (2010) (Fascicule Special)
Tao, T.: The sum-product phenomenon in arbitrary rings. Contrib. Discrete Math. 4(2), 59–82 (2009)
Vaughan, R., Wooley, T.: Waring’s Problem: A Survey, Number Theory for the Millennium, III (Urbana, IL, 2000), pp. 301–340 (2002)
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Covert, D., Demiroğlu Karabulut, Y. & Pakianathan, J. Cayley Digraphs Associated to Arithmetic Groups. Graphs and Combinatorics 35, 393–417 (2019). https://doi.org/10.1007/s00373-018-2002-9
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DOI: https://doi.org/10.1007/s00373-018-2002-9