Tauberian theorems for sum sets

Introduction. The sums formed from the set of non-negative powers of 2 are just the non-negative integers. It is easy to obtain “abelian” results to the effect that if a set is distributed like the powers of 2, then the sum set will be distributed like Dhe non-negative integers. We will be concerned here with converse, or “Tauberian” results. The main theme of this paper is t’he following question: if the set of sums formed from a given set of positive real numbers resembles an arithmetic progression, how much must the original set resemble a set of constant multiples of powers of 2? If we denote the given set by k,, i&, 7c,, . . . , arranged in ascending order, and let S(m) count the number of those sums of distinct ?ci that do not exceed IC, our problem is, roughly, that of showing that k, is close to 2” if S(z) is close to %. Our first result gives sharp bounds for liminf and limsup of 2%/k, in terms of liminf and limsup of S(z)/m. In the next section, we show that if S(X) II: is bounded, then k,2% is bounded, and furthermore, 2 /k,-2finJ ( co, so that if t’he kti are integers, then k, = 2” for all large ‘IL. We extend the method in the suc.ceeding section to obtain estimates for i&2” and 2 [k,2”j in terms of suitable bounds n<N for S(m) z? even if S(m) x: is unbounded. Finally, on a slightly different note, we show t’hat it is not possible for S(a) to behave too much like m” if CL < 1.