What does G(1) = 1 say about a system? This is a line from a paper I've been reading:


  The static gain of the closed loop system must be $1$ ($G(1) = 1$) [...]


First of all: I know what gain is, but isn't gain dependent upon frequency? And why does $G(1)$ do the trick of determining the static gain? And what is "static" gain, anyway? 

We are talking about a discrete-time closed loop system and $G(z)$ is its transfer function.
Static gain refers to the DC gain. Namely, it would be the ratio of the output and the input under steady state condition.

Due to DC corresponding to $\omega=0$, in the $z$-domain DC would correspond to $z=1$ because $z=re^{j\omega}$, with $r=1$ (i.e. you have to evaluate your transfer function $G(z)$ in the unit circle to get the frequency response).

EDIT:

The frequency response of a system is represented in the Fourier domain, $G(\omega)$. The DTFT of a given discrete sequence can be calculated as:

$$G_F(\omega)=\sum_{n=-\infty}^{\infty}g[n]e^{-j\omega n}$$

On the other hand, the $z$-transform is defined as:

$$G_Z(z)=\sum_{n=-\infty}^{\infty}g[n]z^{-n}$$

It's easy to see that if we evaluate $z=e^{j\omega}$ (the unit circle), then the $z$-transform returns the frequency response.