On January 1, 3000, a tachyon beam signal, travelling at four times the speed of light, is sent from Earth towards the starship Tempus Fugitive. The message is "Ping!" At the time the message is sent, the ship is 0.8 light years from Earth, travelling at a speed of 0.8 times the speed of light.
By the time the tachyon signal reaches the starship, it is 1 light year away, as measured from the Earth. But on the Tempus Fugitive, the Earth is only 0.6 light years away (Lorentz contraction).
The date of this event is April 1, 3000, as measured on Earth. But in the reference frame of the starship, this event is contemporaneous with events taking place just after noon on September 3, 2999 on Earth (frame dependence of simultaneity). [Geek alert: t' = gamma*t - L*beta*gamma/c; if t=0.25 years and L=1 l.y., beta=v/c=0.8, and gamma=1/sqrt(1-beta^2)=0.6, then t'=-0.33 years.]
The Tempus Fugitive replies with an "Ack!" upon receipt of the message. It takes .15 years for the signal to traverse that distance, but the Earth is travelling away from the starship at .8 c, so the signal takes .1875 years or 68.4 days for the signal to reach Earth. But in the starship's frame of reference, time on Earth is moving only at .6 its regular speed, so only 41 days pass there (time dilation). The return signal arrives on Earth on October 14, 2999.
This same sort of analysis will work for any signal moving faster than the speed of light. For a signal that is only moving slightly faster than light, the traveller will have to be moving extremely close to the speed of light in order to take advantage of such endochronic properties (egads, another "snide" science fiction reference, Asimov this time), but it can be done in principle. Counterintuitively, the farther away is the traveller, the farther back in time the signal is projected (again, provided he is moving fast enough to do it at all).