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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: Sets of integers that do not contain long arithmetic progressions. | Sets of integers that do not contain long arithmetic progressions Kevin O Bryant Department of Mathematics The City University of New York College of Staten Island and The Graduate Center kevin@member.ams.org Submitted Feb 1 2011 Accepted Mar 2 2010 Published Mar 11 2011 Mathematics Subject Classification 11B25 Abstract Combining ideas of Rankin Elkin Green Wolf we give constructive lower bounds for rk N the largest size of a subset of 1 2 . N that does not contain a k-element arithmetic progression For every e 0 if N is sufficiently large then ro N N I --- fl exp a 8 log N 1 log log N 3 n expiog 4 iog iog en0 2 I V 4 ri. N N Cl. exo n2 n 1 2 a los N 71 los loss N ke Kk expn og 2n ogog where Ck 0 is an unspecified constant log log2 exp x 2x and n flog k . These are currently the best lower bounds for all k and are an improvement over previous lower bounds for all k 4. We denote by rk N the maximum possible size of a subset of 1 2 . N that does not contain k numbers in arithmetic progression. Behrend 1 proved that rsW C exp 1 c ự8log N where exp and log are the base-2 exponential and logarithm and each occurrence of C is a new positive constant. Sixty years later Elkin 2 introduced a new idea to Behrend s work and showed that there are arbitrarily large N satisfying rsN C exp V8 log N 1 log log N This work was supported by a grant from The City University of New York PSC-CUNY Research Award Program. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P59 1 and shortly afterwards Green Wolf 6 arrived at the same bound by a different method. For k 1 2n 1 Rankin 10 proved that for each e 0 if N is sufficiently large then r N C exp n 2 n 1- 2 1 e ựĩogN where n flog k . For k 3 Rankin s construction is the same as that of Behrend. This was subsequently rediscovered in a simpler but less precise form by Laba Lacey 8 . Together with the obvious rk N rk 1 N rk N M rk N rk M these were the thickest known constructions. The primary interest in the current work is the following .
