On A Problem of Erdős and Turán and Some Related Results

Abstract

We employ the probabilistic method to prove a stronger version of a result of Helm, related to a conjecture of Erdős 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]| ⪢ √ x with probability 1, all integers in any interval [ I, N] can be written in at least c 1 log N and at most c 2 log N ways as a difference of elements of A ∩ [1, N 2]. 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) = ∅ 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.