 
Summary: On a problem of Erdos and TurŽan and some related results
Noga Alon
Mihail N. Kolountzakis
March 1995
Abstract
We employ the probabilistic method to prove a stronger version of a result of Helm, related to a
conjecture of Erdos and TurŽan 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 [1, 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) = 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.
Mathematics Subject Classification: 11B13
Notation
