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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Bh Sequences in Higher Dimensions. | Bh Sequences in Higher Dimensions Laurence Rackham Department of Mathematics Royal Holloway University of London Egham Surrey TW20 0EX United Kingdom l.rackham.00@cantab.net Paulius Sarka Department of Mathematics and Informatics Vilnius University Naugarduko 24 Vilnius LT-03225 Lithuania paulius.sarka@gmail.com Submitted Sep 26 2009 Accepted Feb 7 2010 Published Feb 28 2010 Mathematics Subject Classifications 11B05 11B99 Abstract In this article we look at the well-studied upper bounds for A where A c N is a Bh sequence and generalise these to the case where A c Nd. In particular we give d-dimensional analogues to results of Chen Jia Graham and Green. 1 Introduction 1.1 Infinite Bh sequences Let h d G N with h 2. A d-dimensional set A c Nd is called a d-dimensional Bh sequence if all sums a1 a2 ah where a1 a2 . ah G A are different up to rearrangement of summands. We denote A n as number of elements of A in a box 1 n d. If A is a d-dimensional Bh sequence then AhD hn d which implies A n O nd h . 1 Erdos improved this inequality for one-dimensional B2 sequences showing that liminf A n Jỉ gn TO. n V n This result was generalised for d-dimensional B2 sequences by J. Cilleruelo THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R35 1 Theorem 1.1. 1 If A c Nd is a B2 sequence then liminf A nh logn rc. n V nd and for one dimensional B2k sequences by S. Chen Theorem 1.2. 2 If A c N is a B2k sequence then liminf A n klogn rc. n tt y n As noted in 2 no results of this type are known for h odd. 1.2 Finite Bh sequences Erdos and Turan gave the first upper bound for finite B2 sequences showing that if A c 1 N is a B2 sequence then A N1 O N1 . Lindstrom 7 improved the method of this paper to obtain A N1 N4 1. If A c 1 N is a Bh sequence a simple counting argument gives A hh N h. Lindstrom 8 improved this for A c 1 N a B4 sequence proving A 81N4 O N8 . Jia generalised this argument for even h to obtain Theorem 1.3 6 see also 5 . If A c 1 N is a B2k sequence then A k2k k kN2k O N4k
