Generalized Sidon sets of perfect powers

For any fixed integer $k \ge 2$ and an infinite set of positive integers $A$, let $R_{A,k}(n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ distinct terms from $A$. Given positive integers $g \ge 1$, $h \ge 2$, we say a set of positive integers $A$ is a $B_h[g]$ set if every postive integer can be written as the sum of $h$ not necessarily distinct terms from $A$ at most $g$ different ways. A set $A$ of positive integers is called a basis of order $k$ if every positive integer can be written as the sum of $k$ terms from $A$. We say a basis $A$ of order $k$ is thin if the number of representations of $n$ as the sum of $k$ terms from $A$ is positive but small for every positive integer $n$. A few years ago, Van Ha Vu proved the existence of a thin basis of order $k$ formed by perfect powers. In my talk I would like to speak about $B_{h}[g]$ sets formed by perfect powers. I prove the existence of a set $A$ formed by perfect powers with almost possible maximal density such that $R_{A,h}(n)$ is bounded. The proofs are based on the probabilistic method. This is a joint work with Csaba Sándor.
ZOOM meeting ID: 983 9230 2089 Passcode: a six digit number $N=p_4\cdot p_{50}\cdot p_{101}$ where $p_j$ is the j-th prime number.