TAILIEUCHUNG - Báo cáo toán học: "On the number of orthogonal systems in vector spaces over finite fields"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On the number of orthogonal systems in vector spaces over finite fields. | On the number of orthogonal systems in vector spaces over finite fields Le Anh Vinh Mathematics Department Harvard University Cambridge MA 02138 US vinh@ Submitted Jul 15 2008 Accepted Aug 13 2008 Published Aug 25 2008 Mathematics Subject Classification 05C50 05C35 Abstract losevich and Senger 2008 showed that if a subset of the d-dimensional vector space over a finite field is large enough then it contains many k-tuples of mutually orthogonal vectors. In this note we provide a graph theoretic proof of this result. 1 Introduction A classical set of problems in combinatorial geometry deals with the question of whether a sufficiently large subset of Rd zd or Fd contains a given geometric configuration. In a recent paper 3 losevich and Senger showed that a sufficiently large subset oCFd the d-dimensional vector space over the finite field with q elements contains many k-tuple of mutually orthogonal vectors. Using geometric and character sum machinery they proved the following result see 3 for the motivation of this result . Theorem 3 Let E c Fd such that EI Cqd M 1 with a sufficiently large constant C 0 where 0 2 d. Let Afc be the number of k-tuples of k mutually orthogonal vectors in E. Then k 1 o 1 - . k In this note we provide a different proof to this result using graph theoretic methods. The main result of this note is the following. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N32 1 Theorem Let E c Fd such that IE 1 qi 1-1 where d 2 k 1 . Then the number of k-tuples of k mutually orthogonal vectors in E is 1 o 1 11 q- . k Note that Theorem only works in the range d 2 as larger tuples of mutually orthogonal vectors are out of range of the methods uses while Theorem works in a wider range d 2 k 1 . Moreover Theorem is stronger than Theorem in the same range. Sharpness of results It is also interesting to note that the exponent d 1 cannot be improved in the case k 2. In 3 losevich and Senger constructed

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