On A Problem of Erdimages and Turán and Some Related Results Original Research Article

We employ the probabilistic method to prove a stronger version of a result of Helm, related to a conjecture of Erdimages and Turán about additive bases of the positive integers. We show that for a class of random sequences of positive integers A, which satisfy A ∩ [1, x] much greater-than √x with probability 1, all integers in any interval [ I, N] can be written in at least c1 log N and at most c2 log N ways as a difference of elements of A ∩ [1, N2]. We also prove several results related to another result of Helm. We show that for every sequence of positive integers M, with counting function M(x), there is always another sequence of positive integers A such that M ∩ (A − A) = empty set︀ and A(x) > x/(M(x) + 1). We also show that this result is essentially best possible, and we show how to construct a sequence A with A(x) > cx/(M(x) + 1) for which every element of M is represented exactly as many times as we wish as a difference of elements of A.