Integer Sum Sets Containing Long Arithmetic Progressions

the Schnirelmann and lower asymptotic densities respectively of d. According to Schnirelmann theory (see [9]), if 1 > as/ > 0 and Oes/ then a(2s/) ^ 2a(s/)-a(s/) > a(s/); and if a(s/)^\ then 2s/ = No. From this it follows that if as/ > 0 then there exists a positive integer k such that s/ is a basis of order k (that is, ks/ = No). According to Kneser's theorem [6], if dsf > 0 and Oes/ then either s? is an asymptotic basis (in the sense that for some positive integer k, ks/ and No coincide from some point onwards) or some multiple ksf of sf is, from some point on, the union of arithmetic progressions with the same modulus. The outcomes of both theories are best possible, but in neither theory can anything at all be inferred when the relevant density is zero; and even when this density is positive, neither theory is able to describe the progressive enrichment in arithmetical structure (as distinct from density) that occurs under set addition. What we have in mind here is some analogue of Folkman's theorem [4] about partitions. If st is such that A(N) > N for every N > N0(e), then the set 0>(s?) of all integers that can be partitioned into distinct parts from s# contains an infinite arithmetic progression. Intermediate between this result and Theorem 1 below is [1, Lemma 3.9] of Alon and Freiman, which may be viewed as a finite version of Folkman's theorem (sharper versions of this lemma, due respectively to Freiman and Sarkozy, are in course of publication). The object of this note is to establish the following result.