When are all sums of the elements of a set different?

Question

Consider a set $S = \{x_1, \dots, x_n\} \subset \mathbb{Q}\setminus\{0\}$ and assume that for any $I, J \subset [n]$ with $I \neq J$ we have that

\begin{equation} \sum_{i \in I} x_i \neq \sum_{j \in J} x_j \end{equation}

I would like to know if sets of this form have a name (possibly changing $\mathbb{Q}$ for $\mathbb{R}$ or $\mathbb{Z}$), or if there are any known conditions to ensure that $S$ satisfies this condition with some probability.

For example, this holds almost surely for $\mathbb{R}$ if the elements in $S$ are drawn from a continuous distribution, but I am interested in the other cases.

Thanks.

Answer 1

Regarding conditions to decide.

It is NP-complete over the positive integers.

As pointed by Ben Barber, "It does not matter whether you allow $I$ and $J$ to intersect, as replacing $I$ by $I \setminus J$ and $J$ by $J \setminus I$ will preserve equality."

ON THE COMPLEXITY OF VARIATIONS OF EQUAL SUM SUBSETS, p. 1

The Equal Sum Subsets problem, where we are given a set of positive integers and we ask for two nonempty disjoint subsets such that their elements add up to the same total, is known to be NP-hard.

Also related is OEIS A201052 a(n) is the maximal number c of integers that can be chosen from {1,2,...,n} so that all 2^c subsets have distinct sums.