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 Structure of Sets with Few Three-Term Arithmetic Progressions. | On the Structure of Sets with Few Three-Term Arithmetic Progressions Ernie Croot Georgia Institute of Technology School of Mathematics 103 Skiles Atlanta Ga 30332 ecroot@ Submitted Jun 19 2009 Accepted Aug 17 2010 Published Sep 22 2010 Mathematics Subject Classification 11B25 11B30 primary 11N30 secondary Abstract Fix a prime p 3 and a real number 0 a 1. Let S c Fpn be any set with the least number of solutions to x y 2z note that this means that x z y is an arithmetic progression subject to the constraint that S apn. What can one say about the structure of such sets S In this paper we show that they are essentially the union of a small number of cosets of some large-dimensional subspace of F . 1 Introduction Of central importance to the subject of additive combinatorics is that of determining when a subset of the integers 1 . N contains a k-term arithmetic progression. This subject has a long history see 9 ch. 10-11 . In this paper we consider a specific problem in this area posed by B. Green 1 . Before we state this problem we require some notation Given a function f Fpn 0 1 where Fpn denotes the vector space of dimension n over Fp define E f p-raEmein f m . Define A3 f p 2 tm d f m f m d f m 2d . In the case where f is an indicator function for some set S c Fpn we have that A3 f is the normalized count of the number of three-term arithmetic progressions m m d m 2d E S. Supported by NSA grant and NSF grant DMS-1001111. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R128 1 Note that A3 f 0 unless E f 0 because of the contribution of trivial progressions where d 0. Green s problem is as follows Problem. Given 0 a 1 suppose S c Fp satisfies S ap and has the least number of three-term arithmetic progressions. What is A3 S It seems that the only hope of answering a question like this is to understand the structure of these sets S as Green and Sisask did in 5 for values of a near to In this paper we address the analogous problem in F where p is held
