I am wondering if you understand Ohm's law and how it works in this situation. The amplifier needs a certain amount of peak current at a certain voltage. If the voltage goes down, the current demand goes up. Likewise if the voltage goes up the current demand goes down. So in this situation, he is using 4 gauge wire to run BOTH a 1000 watt amp as well as a 4-channel amplifier. So let's say we have 1000 watts at 12 volts, at 100 percent efficiency, you would need 83.3 amps to make this power. Let's add a 4-channel amp at 50 x 4, this amp needs 16 amps to make this power. We have a current demand of approximately 100 amps at 100% efficiency. Now let's get to the real world and say these amps are 85% efficient, now the pair needs 115 amps of current to produce full power. Now keep in mind that this is with all circumstances being perfect, 60 degree wire temp, grounds and power connections and terminations have zero resistance. 4-gauge wire is rated for a continuous 60-80 amps depending on the chart you read and whether or not it is enclosed (covered) which raises temperature. As you reach the capacity of the wire, you will start getting voltage drop. In this case, I can estimate about 1 volt of drop at full draw. Also keep in mind as the temp increases, the current carrying capability of the wire drops and starts heating up the wire which causes further voltage drop, but we can ignore that. Going up to 2 gauge wire adds 20-40 amps of capacity which is 80-120 amps, still just right at or under what is needed for his setup. Being that he won't be listening at full volume 100% of the time, I recommend 2 gauge wire for both current demands and safety concerns. 1/0 would be great, but it is pricey. He can go here and get 20 feet of 2 gauge for less than $40 for power and $6 or so for ground. In my opinion, that is not too much to ask to help a difficult situation. Now on to capacitors. back in the day, they were great since amps did not have on-board capacitance in the quantities that they do now. Adding a capacitor is adding a load to his current system. If he adds a capacitor, it will seem to reduce dimming, but in actuality, it is masking the quick transitions between high and low voltages as the overall average voltage drops due to an undersized charging system. A better choice would be to add a battery to the rear of the vehicle as well, which will initially be a load on the system, but once fully charged will supplement the system. Now if he had a sufficient charging system that could deliver the current he requires and still got dimming, a capacitor may be the correct choice. But, since he has inadequate charging, insufficient main wire capacity as well as insufficient ground wire capacity, that needs to be addressed first.
I have 20+ years of experience in car audio installations with about 16 years of competition installs under my belt. I am also a degreed Certified Electronics Technician with 12 years experience in the field of electronics implementation, service and design. I DID NOT get hot under the collar concerning this issue, just made some sensible suggestions that will help his situation from my experiences.
