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: On the possible orders of a basis for a finite cyclic group. | On the possible orders of a basis for a finite cyclic group Peter Dukes Mathematics and Statistics University of Victoria Victoria BC Canada V8W3R4 dukes@ Peter Hegarty Mathematical Sciences Chalmers University of Technology and University of Gothenburg 41296 Gothenburg Sweden hegarty@ Sarada Herkef Mathematics and Statistics University of Victoria Victoria BC Canada V8W3R4 sarada@ Submitted Oct 5 2009 Accepted May 18 2010 Published May 25 2010 Mathematics Subject Classification 11B13 11B75 primary 05C20 secondary Abstract We prove a result concerning the possible orders of a basis for the cyclic group Zn namely For each k E N there exists a constant ck 0 such that for all n E N if A c Zn is a basis of order greater than n k then the order of A is within ck of n l for some integer l E 1 k . The proof makes use of various results in additive number theory concerning the growth of sumsets. Additionally exact results are summarized for the possible basis orders greater than n 4 and less than ựn. An equivalent problem in graph theory is discussed with applications. Research supported by NSERC Research supported by Swedish Science Research Council Vetenskapsrâdet Research supported by NSERC THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R79 1 1 Introduction Let G be an abelian group written additively and A a subset of G. For a positive integer h we denote by hA the subset of G consisting of all possible sums of h not necessarily distinct elements of A . hA ai ữfo ữị G A . This set is called the h-fold sumset of A. We say that A is a basis for G if hA G for some h G N. Define the function p 2G N x as follows J min h hA G if A is a basis for G p x otherwise. In the case where p A x this invariant is usually referred to as the order1 of the basis A. Now let us specialise to the case G Zn a finite cyclic group. Throughout this paper we will write pn A when referring to a subset A of Zn. Clearly a subset A Zn is a basis if and only if .
