Long arithmetic progressions in sumsets: Thresholds and bounds

Abstract

For a set A A of integers, the sumset l A = A + ⋯ + A lA =A+\dots +A consists of those numbers which can be represented as a sum of l l elements of A A : \[ l A = { a 1 + ⋯ + a l | a i ∈ A i } . lA =\{a_1+\dots + a_l| a_i \in A_i \}. \] Closely related and equally interesting notion is that of l ∗ A l^{\ast }A , which is the collection of numbers which can be represented as a sum of l l different elements of A A : \[ l ∗ A = { a 1 + ⋯ + a l | a i ∈ A i , a i ≠ a j } . l^{\ast }A =\{a_1+\dots + a_l| a_i \in A_i, a_i \neq a_j \}. \] The goal of this paper is to investigate the structure of l A lA and l ∗ A l^{\ast }A , where A A is a subset of { 1 , 2 , … , n } \{1,2, \dots , n\} . As application, we solve two conjectures by Erdös and Folkman, posed in 1960s.