Does there exists a theorem like this? Statement: Suppose $T\subseteq \mathbb{N}$, then all $x^i,i\in T$ generate a dense linear subspace of $C^0[a,b]$ iff $\sum_{i\in T} 1/i$ is divergent.
I heard it somewhere a long time ago, so there may be minor errors, but the meaning goes like this. I heard it was called a "Bernstein problem", but I never succeeded in searching for such a theorem on the web.

An excellent reference, not mentioned in the Wikipedia article, is section 4.2 of Polynomials and Polynomial Inequalities by Borwein and Erdélyi. On 35 pages of that section the authors collect a huge number of variations of the theorem (and then return to it in 4.4 in the setting of rational functions).
Here is a sample.
Theorem 4.2.1  Suppose $(\lambda_n)$ is a sequence of distinct positive numbers. Then the span of $x^{\lambda_n}$ is dense in $C[0,1]$ if and only if 
$$\sum_{n} \frac{\lambda_n}{\lambda_n^2+1}=\infty \tag1$$
Condition (1) simplifies to $\sum_n \lambda_n^{-1}=\infty$ when $\inf \lambda_n>0$.
And a rational version:
Theorem 4.4.1 Let $(\lambda_n)$ be any sequence of distinct real numbers. Then, for any $0<a<b$,  the set 
$$\left\{ \frac{\sum_{n=0}^N a_n x^{\lambda_n}}{\sum_{n=0}^N b_n x^{\lambda_n}}  : a_n,b_n\in\mathbb R, \ N=1,2,3\dots \right\}$$
is dense in $C[a,b]$. 
