What does the ratio $\frac{x-ct}{x'-ct'}$ signify? $(x,t)$ being coordinates of any event in an inertial frame $A$ and $(x',t')$ being the coordinates of the same event in another frame moving with uniform velocity $v$ along the same direction in which $x$ or $x'$ have been measured (when $t=0$ and $x=0$, $x'=0$ and $t'=0$). The ratio
$$\frac{x-ct}{x'-ct'} = \sqrt{\frac{1-v/c}{1+v/c}}$$
Being the same ratio as it comes out in Doppler effect of light, I thought it might suggest something significant. (Perhaps something about the properties of spatial distance between the light pulse sent in positive $x$ direction at $t=0$ from $x=0$ (and thus at $t'=0$ and $x'=0$) and the spatial position of the event concerned).
 A: Let's first take a look at what $x-ct$ means. Up to a phase, a light plane-wave can be written as
$$\sim A^\mu \exp[ik_\nu x^\nu] \equiv A^\mu \exp[ik(x-ct)] \equiv A^\mu \exp[i\omega(x/c-t)]$$
I.e., along $x-ct=\rm const.$ the phase of the wave is constant. Hence we can define $x-ct \equiv \Delta \kappa(x,t)$ as a parametrization of "phase-time" along a light-wave with respect to the coordinate time and coordinate distance. ($\Delta \kappa$ has in fact the dimension of distance, but "phase-time" sounds better.)
The ratio 
$$\frac{x-ct}{x'-ct'} = \frac{\Delta \kappa}{\Delta \kappa'}$$ 
is then simply the ratio of the phase-times of the lightwaves as measured in the respective inertial frames. 
Now consider the following, $k_\nu x^\nu \equiv k_0 ct + k_x x \equiv -\omega t + k x = k \Delta \kappa = \omega \Delta \kappa /c$ is the scalar product of two four-vectors $k^\mu$ and $x^\mu$ and hence a relativistic invariant. This means that $\omega \Delta \kappa=\omega' \Delta \kappa'$ under any Lorentz transformation. Another way to state this is
$$ \frac{\Delta \kappa}{\Delta \kappa'} = \frac{\omega'}{\omega} $$
But $\omega'/\omega$ is the Doppler shift in frequency of light and the correspondence with the "phase ratio" is now clear. (But beware, depending on the situation one might confuse the Doppler shift and it's inverse!)

Just a side note, the coordinates $U=(x-ct)/\sqrt{2},\, V=(x+ct)/\sqrt{2}$ are often used to simplify problems involving propagation at the speed of light because under the right construction the light-waves then depend either only on $U$ or $V \!$.
