Generalized Sidon sets of perfect powers

Abstract

AbstractFor $$h \ge 2$$ h ≥ 2 and an infinite set of positive integers A, let $$R_{A,h}(n)$$ 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 $$B_h[g]$$ 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 $$B_{h}[g]$$ 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 $$R_{A,h}(n)$$ R A , h ( n ) is bounded by using probabilistic methods.