"is there any simple fix for this?"

The fix is possibly simple,finding which of the dozen possible simple fixs it needs is what separate's men from boys. Seriously,call an electrician before something bad happens.

"Will adding a large capacity surge protector where the AC is plugged in do any good?"

It will not do any good. Surge protector protect's equipment against surges of higher than normal volts. Your ac is suffering from surges of LOWER than normal volts.

Who is your cable company and what model set top box do you have?

Call your cable company and tell them you cable box is having problems. Ask them to give you a new one., preferably another model. A set top box has a power on reset function. What is probably happening is the box sees a drop in voltage and generating a reset. It is hard to believe the voltage is dropping low and long enough for the box's reset circuit to detect this, but I suppose it could happen. Perhaps another set box model would be less sensitive to the momentary voltage drop.

If swapping boxes doesn't work then you may want to invest in a UPS supply for $40 on Amazon. This device has a built battery back up which should power the set top box through any drop in voltage. There are several of these available on Amazon.

Calling a cable company or installing a UPS is a very seriously irresponsible answer. That dimming indicates a potentially hazardous wiring problem. Maybe just badly connected wires. In some cases, a potential house fire.

One who does not know simple electrical concepts has no business making recommendations.

Why would anyone be so irresponsible as to cure symptoms with a UPS? If wiring is safe, then no dimming occurs. If voltage is dropping so massively as to even reset a cable box, then dimming indicates a defect that is dangerous.

Dimming also says a defect is potentially harmful to the air conditioner. Motorized appliances dislike a low voltage on startup. That low voltage will cause early air conditioner failure.

Quite accurate is a reply that says a surge protector will also do nothing useful. Every adult should know this layman simple concept. A 120 volt surge protector does nothing until voltage rises well above 330 volts. Obviously voltages is dropping to well below 120 volts (probably less than 85 volts). At no time does voltage rise above 330 volts. So a protector does nothing - remains completely inert.

The OP never used the word "dimming" in his original post. He also stated the set top box will reset itself when the AC is running. So it is difficult to determine if the root cause is the house wiring and if is a serious electrical problem. From the brief description it sounded more like a set top box issue than a wiring problem. But perhaps you are correct and the circuit breaker, outlet receptacle, and the wiring have to be replaced. I take accusations of irresponsibility seriously especially when it comes to electrical wiring. Perhaps I should have asked more questions before offering a solution.

After everything is replaced I would not be shocked if the problem still occurs. In recent years manufactures have been pressed to cost reduce the price of their boxes. An older box with a more robust AC power supply with big capacitors may have ignored a voltage spike, but a newer box could be susceptible to low voltage transients.

Correct. He did not use the word 'dimming'. What he did say (and I forgot to mention this) is made obvious by an incandescent bulb powered from the same circuit. A wiring defect will cause voltage to drop (what he called 'power to dip'). If that voltage drop is massive, then even a router will reset. To reset a router requires a major voltage drop. Put numbers to it: a router will reset if 120 volts drops below 85.

More numbers; that would probably cause an incandescent bulb to dim to less than 40% intensity. If any motorized appliances causes any noticeable dimming in an incandescent bulb, then a workmanship defect probably exists. If it dims to 50% or less, then that defect is probably quite serious. His defect is so bad that even a router is resetting.

Most important: 'magic box' solutions are bogus. Do not cure symptoms. Always cure a defect.

Today's electronics are happy even with greater voltage variations compared to yesterday's. Specs have advanced every decade. For example, electronics would power off if voltage dropped so low that a bulb dimmed to 50% intensity. Today's computers must be perfectly happy even when that bulb dims to 40% intensity. Despite popular fables, today electronics operate uninterrupted with even greater voltage variations.

Unfortunately, too many are educated by myths, lies, hearsay, and wild speculation that promotes a UPS. The informed know a UPS is 'dirty' and temporary power to protect unsaved data. UPS does nothing to (and does not even claim to) protect hardware. How does one know? Responsible recommendations always include numbers.

No number justifies a UPS. No number says today's electronics are more sensitive to voltage variations. But that is what naive consumers are told to believe by hearsay, wild speculation, salesmen, advertising, and subjective reasoning. A subjective recommendation (no numbers) indicates that it is best ignored. Honest and informed recommendations also provide numbers.

A responsible recommendation called for an electrician as soon as possible to fix 'lower than normal' voltage. In most cases, that defect will not threaten human life. In rare cases, it does; should not be ignored.