Let $\Lambda :=\{\lambda_1, \dots, \lambda_n\}$ be a set of $n$ distinct real numbers.

For a given $p \in \mathbb N$, consider further the set

$$I_p := \{ \{i_1, i_2, \dots, i_p\} : i_j \in \{1, \dots,n\} \text{ for all } j=1, \dots, p\}.$$

It is an easy and well-known fact that $|I_p| = \binom{n+p-1}{p}$. For example, if $n=3$ and $p=2$, then

$$ I_2 = \{ \{1,1\}, \{1,2\}, \{1,3\}, \{2,2\}, \{2,3\}, \{3,3\} \}.$$

Now for a given $p \in \mathbb N$ we write down all $\binom{n+p-1}{p}$ sums $$\lambda_{i_1} + \lambda_{i_2} + \dots + \lambda_{i_p}. $$

I was asking myself if there was a (rather) simple condition on the set $\Lambda$ such that one can guarantee that for all $p \in \mathbb N$ all the $\binom{n+p-1}{p}$ sums will be distinct, i.e. a condition on $\Lambda$ such that

$$ \lambda_1, \dots, \lambda_n \text{ pairwise distinct and also}$$ $$ \lambda_1 + \lambda_1, \lambda_1 + \lambda_2, \dots, \lambda_1+\lambda_n, \lambda_2 + \lambda_2, \dots \text{ pairwise distinct and also...}$$ $$ \text{etc.}$$