My example is already an extremely progressive property tax on multiple ownership, but yeah, it can be tailored as needed.

I, personlly, would be in favor of doubling the tax if no one is living there.

As for rent: the price of rent cannot be arbitrarily raised. If renters can buy more cheaply than renting, then landlords will have empty units.

So in your example, renters would just buy and pay the $3500 mortgage instead of renting.

Of course real life is more complex. Renters need access to financing, etc.