Consecutive Integers in High-multiplicity Sumsets

Sharpening (a particular case of) a result of Szemerédi and Vu [SV06] and extending earlier results of Sárközy [S89] and ourselves [L97b], we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset to contain a block of consecutive integers of length, comparable with the lengths of the set summands. A corollary of our main result is as follows. Let k, l > 1 and n > 3 be integers, and suppose that A1, . . . , Ak ⊆ [0, l] are integer sets of size at least n, none of which is contained in an arithmetic progression with difference greater than 1. If k > 2 ⌈(l − 1)/(n − 2)⌉, then the sumset A1+· · ·+Ak contains a block of consecutive integers of length k(n − 1). 1. Background and summary of results The sumset of subsets A1, . . . , Ak of an additively written group is defined by A1 + · · ·+ Ak := {a1 + · · ·+ ak : a1 ∈ A1, . . . , ak ∈ Ak}; if A1 = · · · = Ak = A, this is commonly abbreviated as kA. In the present paper we will be concerned exclusively with the group of integers, in which case a well-known phenomenon occurs: if all sets Ai are dense, and their number k is large, then the sumset A1 + · · · + Ak contains long arithmetic progressions. There are numerous ways to specialize this statement, indicating the exact meaning of “dense”, “large”, and “long”, but in our present context the following result of Sárközy is the origin of things. Theorem 1 (Sárközy [S89, Theorem 1]). Let l > n > 2 be integers, and write κ := ⌈(l + 1)/(n− 1)⌉. Then, for every integer set A ⊆ [1, l] with |A| = n, there exist positive integers d 6 κ − 1 and k < 118κ such that the sumset kA contains l consecutive multiples of d. In [L97b] we established a sharp version of this result, replacing the factor 118 with 2 (which is the best possible value, as conjectured by Sárközy) and indeed, going somewhat further. The author gratefully acknowledges the support of the Georgia Institute of Technology and the Fields Institute, which he was visiting while conducting his research. 1