The Sum of the Reciprocals of a Set of Integers with No Arithmetic Progression of Á: Terms1

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 2„6Si \/n > (1 e)k log A:, 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 &(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. Erdös and Davenport [1] proved that for any such Sk, liminf \Sk c\[l,n]\/n = 0, where \X\ denotes the cardinality of X, and [1, n] the set of integers from 1 to n inclusive. More recently, Szemerédi [2] proved that lim \Sk n[l,/il|//i = 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, \Sk n[l,«]| > /iexp[-c(log«)Ä] where b and c are positive numbers which depend on k but not on n. These results have led Erdös [6] to conjecture that 2„es \/n must converge. If this conjecture is true, then for each k > 3, there exists Ak = sup 2 !/"• 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 10L10; Secondary 10H20.