Permutations of the positive integers with restrictions on the sequence of differences, II

In this paper we discuss the following conjecture: Conjecture: Let D = (D₁, … , D_n), D ⊂ N, N the set of positive integers. Then there exists a permutation of N, call it (a_k: k ϵ N) such that (|αf_k+1 − a_k| : k ϵ N) = D iff (D₁, …, D_n) = l. We also consider the following question: Question: For what sets D = (D₁, ‖, D_n) does there exist an integer M ϵ N and a permutation (|b_k:+1: k = 1,… , M) of (1, …, M) such that (|b_k+1 − b_k|: k = 1, …, M - l) = D. We answer the conjecture and the following question in the affirmative if the set D has the following property: For each D_rEspilon; D there is a D_sϵ D such that (D_r, D_s) = 1.