Given $C \geq 1$ and $\epsilon > 0$, is there a number $N = N(C,\, \epsilon)$ such that the following holds:
For every set $S \subseteq S^1$ of cardinality $C$, there is a function $f: S^1 \to \mathbb{R}$ such that:
1). $f(z) = \sum_{n = -N}^N a_n z^n$ (i.e. $f$ is a Laurent polynomial of in degrees $[-N,\, N]$).
2). $f \geq 0$ on $S^1$
3). $f(z) \geq 1$ for each $z \in S$
4). $\|f\|_{L^1} < \epsilon$