# How to find sets of polynomially bounded numbers whose subset sums are different?

Let $$n$$ be any positibe integer and set $$N=\{1,\dots,n\}$$. Now select two arbitrary but different subsets of $$N$$, say $$S$$ and $$S'$$. We are interested in finding a function $$\pi(A)=\sum_{i\in A}a_i$$ such that we always have $$\pi(S) \ne \pi(S')$$. For example, we can define $$a_i = 2^i$$. This function satisfies the property because for each $$A \subseteq N$$, $$\pi(A)$$ uniquely represents an integer written in base 2. However, in this function $$a_i$$ is bounded above by $$\mathcal{O}(2^n)$$. I am looking for a function such that $$|a_i|$$ and $$\frac{1}{|a_i|}$$ are bounded above by a polynomial. As an example that does not satisfy this last condition we can consider $$a_i = 2^{i-n}$$. It would be great to know different functions that satisfy this property. I will try to proof the uniqueness myself (if it is not so difficult!), so please let me know if you are aware of such functions.

• You can choose $n$ random numbers between $0$ and $1$, and almost surely all sums will be different. If you want an explicit example, you can take $a_i = \pi^i/\lceil \pi^i \rceil$. – Yuval Filmus Oct 25 '18 at 4:09
• I'm not writing any answer, you can go ahead. – Yuval Filmus Oct 25 '18 at 5:32

1. $$a_i=1+c^{-i}$$, for some constant $$c\ge 2$$.
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 $$p_i-1$$.
3. $$a_i =1+\alpha^i/\lceil\alpha^i\rceil$$, 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)
4. $$a_i=c_1b_i+c_0$$, where $$b_i>0$$ is such a function and $$c_1, c_0$$ are positive constant.
5. 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.