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- W2012104750 abstract "It is shown that for each integer k > 3, there exists a set Sk of positive integers containing no arithmetic progression of k terms, such that neSk l/n > (1 e)k log k, with a finite number of exceptional k for each real e > 0. This result is shown to be superior to that attainable with other sets in the literature, in particular Rankin's sets 6(k), which have the highest known asymptotic density for sets of positive integers containing no arithmetic progression of k terms. Let Sk be any set of positive integers which contains no arithmetic progression of k terms. Erdos and Davenport [1] proved that for any such Sk, liminf ISk n [1,n] l/n = 0 where IXI denotes the cardinality of X, and [1, n] the set of integers from I to n inclusive. More recently, Szemeredi [2] proved that lim iSk n[1, n]j/n = 0. On the other hand, it has been shown by Behrend [3] and Moser [4] in the case k = 3, and by Rankin [5] for all k > 3, that there exist sets Sk, with no arithmetic progression of k terms, such that, for all positive integers n, ISk n [1, n]I > n exp[-c(logn)b] where b and c are positive numbers which depend on k but not on n. These results have led Erdos [6] to conjecture that E IESk /n must converge. If this conjecture is true, then for each k > 3, there exists Ak =sup l/n. all Sk n E Sk For suppose Ak did not exist for some k. Let Sk(l) be any set of positive integers containing no arithmetic progression of k terms, and, for each integer m > 1, let ak(m) be the least integer such that Received by the editors March 22, 1976. AMS (MOS) subject classifications (1970). Primary IOL0; Secondary 10H20." @default.
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- W2012104750 date "1977-02-01" @default.
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- W2012104750 title "The sum of the reciprocals of a set of integers with no arithmetic progression of $k$ terms" @default.
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- W2012104750 doi "https://doi.org/10.1090/s0002-9939-1977-0439796-9" @default.
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