Generalized Sidon sets of perfect powers

For h ≥ 2 and an infinite set of positive integers A , let R A , 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 , h ( n ) is bounded by using probabilistic methods.