Consecutive integers in high-multiplicity sumsets

Abstract

Abstract. 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 A 1 , . . . , A k ⊆ [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 A 1 +· · ·+A k contains a block of consecutive integers of length k(n − 1).

Sufficient conditions for perfect mixed tilings

“…We require the following result. Lemma 6.3 (Lev [25]). Suppose that s 1, n 3 are integers and A 1 , .…”

“…In fact, whether a collection of graphs from F cr forms a flexi-chromatic graph quickly reveals itself to be a problem in additive combinatorics; it boils down to the question of whether the set of all potential sums of a certain set of integers contains a long interval. In this paper we adopt a result of Lev [25] to our setting. Questions of this type have recently been investigated by Conlon, Fox and Pham in [6].…”

Subset sums, completeness and colorings

“…We will also make repeated use of the following result of Lev [30]. The importance of this result is that it allows us to find long intervals in a set of subset sums by first finding several dense subsets of long intervals and then summing these sets.…”

“…Several weaker versions of this result appeared earlier in the literature, many of which would also suffice for our purposes. Lemma 2.2 (Lev [30]). Suppose ℓ, q ≥ 1 and n ≥ 3 are integers with ℓ ≥ 2⌈(q − 1)/(n − 2)⌉.…”