TITLE: Characterizing sets with a function
QUESTION [5 upvotes]: Given a set $A=\{z_1,\ldots,z_n\},\epsilon>0$ where $z_i \in \mathbb{N}$, I wonder if there is an easy way to choose numbers $0<\lambda_1,\ldots,\lambda_n<\epsilon$ explicitly such that for all subsets $T\subseteq\{1,\ldots,n\}$ we have that the numbers
$$f(T)=\sum_{t \in T}\sqrt{z_{t}+\lambda_{t}}$$
are pairwise different? That is $f(T)\neq f(T')$ if $T \neq T'$.
The square root seems to make things difficult, otherwise one could choose something like $\lambda_i=2^{-i -k}$ for some constant $k>0$.

REPLY [2 votes]: Let $$\lambda_{t}=2\sqrt{z_{t}}2^{-i-k}+2^{-2i-2k}$$
 where $k>0$
  is some constant. Then for $k$
  large enough, we have that $0<\lambda_{t}<\epsilon.$
 However, $$\sqrt{z_{t}+\lambda_{t}}=\sqrt{z_{t}}+2^{-t-k},$$
 and so $$\sum_{t\in T}\sqrt{z_{t}+\lambda_{t}}=\sum_{t\in T}\sqrt{z_{t}}+\sum_{t\in T}2^{-t-k}.$$ The above can be viewed as two sums, $\Sigma_1$ and $\Sigma_2$. The idea is that if we choose $k$ large enough, then $\Sigma_1$ will not interact with $\Sigma_2$, and $\Sigma_2$ guarantees uniqueness. 
 Let $$\kappa=\min_{T_{1}\neq T_{2}}\left\{ \sum_{t\in T_{1}}\sqrt{z_{t}}-\sum_{t\in T_{2}}\sqrt{z_{t}}\neq0\right\} ,$$
 that is let $\kappa$ be the minimum nonzero distance between different $\Sigma_1$'s, and then choose $k$
  to be so large that $2^{-k/2}\leq\kappa.$
 Then it follows that the sums $$\sum_{t\in T}\sqrt{z_{t}}+\sum_{t\in T}2^{-t-k}$$
 will be unique for any $T\subset\{1,\dots,n\}.$
 .