TAILIEUCHUNG - Báo cáo toán học: "Sets of integers that do not contain long arithmetic progression"

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@ 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 .

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