Long arithmetic progressions in sumsets: Thresholds and bounds

For a set A A of integers, the sumset l A = A + ⋯ + A lA =A+ +A consists of those numbers which can be represented as a sum of l l elements of A A : \[ l A = { a 1 + ⋯ + a l | a i ∈ A i } . lA =\{a_1+ + a_l| a_i ∈ A_i \}. \] Closely related and equally interesting notion is that of l ∗ A lA , 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 } .