Tax proportionately to the footprint of the property, growing non-linearly.
Tax non-primary-residence units at at least 5X the rate; tax unoccupied units at 10X the rate.
Then there's an incentive to build up, while not screwing over existing residents.
Property tax / land value tax on primary residence is just rent collected by the government for living somewhere.
That rent shouldn't be higher in lucrative locations. Grocery store employees should be able to live where they work; we should not be pushing them away with higher taxes on top of the pressure the market puts on them.
Perhaps food and shelter shouldn't be taxed altogether. The government shouldn't be collecting a rent on people existing.
That would also make redlining of residential neighborhoods harder, as if will force decoupling of local infrastructure funding from property values.
Still much better than status quo.
In the end it approximates land value tax, is more politically feasible, so it is likely a step in the right direction.
Shell company of a dog doesn't have a primary residence.
As for the rest, either they'll have to live there (in which case, it's fair), or they'll go behind the bars for properly tax evasion.
And it doesn't take much to find out that a property listed as primary residence for tax purposes is being rented out. Add some reporting incentives, and hey, that loophole takes care of itself.
Nice idea, but has loopholes.