tailieunhanh - Báo cáo toán học: "On the Structure of Sets with Few Three-Term Arithmetic Progressions"

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