Đang chuẩn bị nút TẢI XUỐNG, xin hãy chờ
Tải xuống
Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Orthogonal systems in vector spaces over finite fields. | Orthogonal systems in vector spaces over finite fields Alex losevich and Steven Senger Department of Mathematics University of Missouri Columbia MO 65211-4100 iosevich@math.missouri.edu senger@math.missouri.edu Submitted Jul 24 2008 Accepted Dec 2 2008 Published Dec 9 2008 Mathematics Subject Classifications 11T23 05B15 Abstract We prove 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. Contents 1 Introduction 1 1.1 Graph theoretic interpretation. 3 1.2 Hyperplane discrepancy problem . 3 1.3 Acknowledgements. 3 2 Proof of Theorem 1.1 3 3 Sharpness examples 8 1 Introduction A classical set of problems in combinatorial geometry deals with the question of whether a sufficiently large subset of Rd Zd or F contains a given geometric configuration. For example a classical result due to Furstenberg Katznelson and Weiss 5 see also 2 says that if E c R2 has positive upper Lebesgue density then for any Ỗ 0 the Ỗ-neighborhood of E contains a congruent copy of a sufficiently large dilate of every three point configuration. When the size of the point set is smaller than the dimension of ambient Euclidean space taking a -neighborhood is not necessary as shown by Bourgain in 2 . He proves A. losevich was supported by the NSF Grant DMS04-56306 and S. Senger was supported by the NSF Grant DMS07-04216 THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R151 1 that if E c Rd has positive upper density and A is a k-simplex with k d then E contains a rotated and translated image of every large dilate of A. The case k d and k d 1 remain open however. See also for example 3 4 9 14 and 16 on related problems and their connections with discrete analogs. In the geometry of the integer lattice Zd related problems have been recently investigated by Akos Magyar in 12 and 13 . In particular he proves in 13 that if d 2k 4 and E c Zd has positive upper density then all large depending on density of E .
