Abstract
If A is a finite set of positive integers, letE_h(A) denote the set of h-fold sums andh -fold products of elements of A. This paper is concerned with the behavior of the function f_h(k), the minimum of E_h(A) taken over all A withA=k . Upper and lower bounds for f_h(k) are proved, improving bounds given by Erdős, Szemerédi, and Nathanson. Moreover, the lower bound holds when we allow A to be a finite set of arbitrary positive real numbers.
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