AN IMPROVEMENT OF THE CONDITION FOR A SET OF POSITIVE INTEGERS TO BE SUBSET-SUM-DISTINCT

Abstract

Let $S$ be a set of positive integers. Then we say that $S$ is a subset-sumdistinct set (briefly, $S$ is an SSD-set) if for any two finite subsets $X, Y$ of $S$,

$$
\sum_{x \in X} x=\sum_{y \in Y} y \Rightarrow X=Y
$$

For fixed positive integers $n$ and $p$, we set

$$
S(n, p):=\left\{1^p, 2^p, 3^p, \ldots, n^p\right\} .
$$

In our previous paper, we proved that $S(n, p)$ is an SSD-set when $p \geq n$. In this paper, we improve the bound as $p \geq 0.71 n$.

Received: August 11, 2025
Accepted: September 1, 2025