TAILIEUCHUNG - Báo cáo toán học: "Bh Sequences in Higher Dimensions"

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 Paulius Sarka Department of Mathematics and Informatics Vilnius University Naugarduko 24 Vilnius LT-03225 Lithuania 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 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 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 . 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. 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 6 see also 5 . If A c 1 N is a B2k sequence then A k2k k kN2k O N4k

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