Had a call years back that the lights in the house would get brighter when the a/c came on. I asked to verify that they were getting brighter, not dimmer, the reply was brighter.
Asked if it could wait a couple of days, answered yes. Next day call said there was arcing behind his electric stove.
Went out. The range was electric, but had gas ran to range location, in-case they ever wanted to change. Arcing was occurring, and the gas pipe had pits in it due to the arcing. Checked the outside main panel & found very loose neutral (had used the ground strands from the outer wrapping of SEU cable for the neutral). Tightened wires, problem fixed.
A 240 volt AC unit would not cause this as it does not put a load on the neutral, the blower motor in the furnace or air handler would though which is what they might have been seeing, as for the range older electric ranges did use the neutral for some of the smaller top burners or when they were set to lower settings, but most today use only the two hots and switch between different elements to control the amount of heat, it sounds like the range only had a 3-wire hookup and was using the neutral for both the EGC and neutral as was allowed in older installations.
The gas pipe must have had a electrical path to another service and would have had to be all metal which is very rare today, because if it was only connected to this houses grounding/neutral it would have been at the same potential of this houses neutral and there would have been no arcs, it could have been through a water heater to the water pipe which would make sense then the water pipe would have made the connection to the street main which is connected to other service neutrals.
I had a house that had a 30 amp 120 volt only service, no 240 volts, the customer had three different electricians out to find out why the cable TV's wire was burning up, at one time it actually caught fire, over time they had put all 30 amp fuses in the 4 space fuse panel so the load was more around 60 amps at times when they were using the microwave and toaster and coffee pot along with everything else in the house, the other electricians didn't have a clue as they told the owner they never ran into something like this, I just shook my head when she told me this and I showed her where a tree limb had fallen and hit the drop pulling the neutral apart leaving the hot still connected, I just used a split bolt to put the neutral back together and all was well, I also told her that the house was in serious need of updating and that having the 30 amp fuses in the panel was a big fire hazard just waiting to happen, but she didn't have much money and I had to go to another job she said she would have her son put 20 amp fuses in it as all the circuits (all 4) were in #12, it's amazing how people survive like this.