Consecutive integers in high-multiplicity sumsets

TL;DR

This paper establishes a sharp threshold for the number of integer sets needed so that their sumset contains a long block of consecutive integers, extending previous results and sharpening known theorems in additive combinatorics.

Contribution

It provides a new, precise threshold for the number of sets required for their sumset to contain a large block of consecutive integers, extending earlier work of Szemeredi, Vu, Sarkozy, and the authors.

Findings

Identifies a sharp threshold for sumsets to contain long consecutive blocks.

Extends previous results by sharpening conditions and thresholds.

Provides explicit bounds relating set sizes, number of sets, and length of consecutive blocks.

Abstract

Sharpening (a particular case of) a result of Szemeredi and Vu and extending earlier results of Sarkozy and ourselves, 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 $k,l\ge 1$ and $n\ge 3$ be integers, and suppose that $A_1,...,A_k\subset[0,l]$ are integer sets of size at least $n$, none of which is contained in an arithmetic progression with difference greater than 1. If $k\ge 2\lceil(l-1)/(n-2)\rceil$, then the sumset $A_1+...+A_k$ contains a block of consecutive integers of length $k(n-1)$.