Page 3 of this doc [1] lists the definition of city by state

Apparently most states recognize a place as a city if it just has a few hundred residents/voters. There's plenty of abandoned houses (thus $0 homes) in places that fit these definitions.

[1] https://www2.census.gov/geo/pdfs/reference/GARM/Ch9GARM.pdf

Okay, but it's still likely uninteresting because of the Sorites paradox. There's an arbitrary definition of when a town becomes a city because there has to be some threshold, and the list of cheapest cities is probably mostly indistinguishable from a list of cities that were the closest to being towns.

What would be interesting (to me anyway) is a list of some cities who are disproportionately cheap related to their density.