[yes_really]

I think it depends. In the US, denser cities are more expensive than lower density ones. If a city doubles in population, it will probably generate more demand due to the extra jobs opportunities/amenities/culture/etc, so people from other parts of the country will want to move there.

The exception is if everywhere in the country increases in size at the same time. For example, Japan has cheap rent at the most populous city at Earth (Tokyo). But one quarter of the whole country lives there.

If the US had something equivalent, with 80 million people in NYC, and plenty of high-rises and missing middle in every city that is currently mostly single-family-homes, then we could have cheap rents everywhere. But if just one city increases density alone, it seems it will just attract people from elsewhere in the country