How to find sets of polynomially bounded numbers whose subset sums are different?
There are many functions that satisfy your condition. Here are a few.


$a_i=1+c^{-i}$, for some constant $c\ge 2$.
$a_i= 1+b_i/p_i$, where $p_i$ is the $i^{th}$ prime number and $b_i$ is any integer between 1 and $p_i-1$.
$a_i =1+\alpha^i/\lceil\alpha^i\rceil$, for any positive transdental number $\alpha$. For example, you can take $\alpha = \pi$ or $\alpha=e$ or $\alpha=0.110001000000000000000001\cdots$, an Liouville number. (Yuval's method)
$a_i=c_1b_i+c_0$, where $b_i&gt;0$ is such a function and $c_1, c_0$ are positive constant.
One can choose random numbers between a positive constant $c_0$ and 1, which should satisfy the requirement with probability one. The issue then is how to select/implement a random number generator and detect the outliers, which may or may not be trivial. (Yuval's method)


For the first three methods, the constant 1 can be replaced by some kind of polynomial of $n$. For example, a nonzero polynomial of $n$ with nonnegative integer coefficients.

These functions should get you going. If you find more ways, you can update this answer.