Integer Sum Sets Containing Long Arithmetic Progressions

Let jrf, &,... denote subsets of the set No of all the non-negative integers and let denote the sum set {a + b: aesf,be@l}. Write 2s / for s / + s / and, more generally, ksf for sf + (k—\\)sf (fc = 2,3,...). For any positive integer N let A(N): = #(jaf n [l,N]) be the counting number of sf, and