Generalized Sidon sets of perfect powers

For h >= 2 and an infinite set of positive integers A, let R-A,R-h(n) denote the number of representations of the positive integer n as the sum of h distinct terms from A. A set of positive integers A is called a B-h[g] set if every positive integer can be written as the sum of h not necessarily distinct terms from A at most g different ways. We say a set A is a basis of order h if every positive integer can be represented as the sum of h terms from A. Recently, Vu [17] proved the existence of a thin basis of order h formed by perfect powers. In this paper, we study weak B-h[g] sets formed by perfect powers. In particular, we prove the existence of a set A formed by perfect powers with almost possible maximal density such that R-A,R-h(n) is bounded by using probabilistic methods.