Let $\mathbb{A}$ be a set and $\mathbb{A}=\{r_1,r_2,\ldots, r_k\}$ where $r_1,r_2,\ldots,r_k$ are distinct, non-zero reals. Let's say that the sum of a subset $\mathbb{S}$ of $\mathbb{A}$ is the sum of all elements of this subset $\mathbb{S}$.
Is there a nice, non-trivial upper bound on the number of such subsets of $\mathbb{S}$ that have distinct sum?
In general the upper bound is $2^k$.
To see this choose $\mathbb{A} = \{10^i: 0\leq i < k\} = \{1,10^1,10^2,\ldots,10^k\} $. Any sum of a subset will just be a unique binary sequence with k elements, thus we have $2^k$ different such.