> Maybe exclude property that’s actually lived in?

Alas, that would be a terribly idea, because all of a sudden you have the government snooping into your living arrangements. Especially for grey areas where people share their time between multiple apartments. Way too complicated.

And, people living in bigger places would get effectively get a discount this way.

Your argument does highlight a problem with the transition phase of the introduction of the tax, though. There's a few ways to solve this, eg by slowly phasing the tax in over a decade or more.

(It's not a problem in the long run, because land prices will drop until the financial burden of tax plus mortgage payment is the same as today.)

"Alas, that would be a terribly idea, because all of a sudden you have the government snooping into your living arrangements. Especially for grey areas where people share their time between multiple apartments. Way too complicated."

... not really. Most folks only own one house, so nothing complicated. A few folks own multiple houses. These folks already need to establish residency in one house for tax purposes in a lot of places: Taxes vary from city to city and state to state. It isn't a new grey area, and we can use this established system to tax. The folks can, in general, just use one house or apartment or pay more for more than one house.

There's still extra complexity for renting. (Unless you want to not give the discount to places that renters live in?)

Not really any more than you already have. In Indiana, they already tax folks higher for homes you don't personally live in (no discounts here). Not only that, but rental income is income: This is a business you are running, after all, and you already have to file taxes on that income (and expenses). It truly wouldn't be that difficult to register a house as a rental and giving it the discount if you think it should apply.

I'd personally only give it out to folks with longer terms: The place should still be someone's primary residence, and only give a discount if it has been rented out x months out of the year. This would be to discourage airBNB units (for example) and encourage landlords to make their units affordable and attractive to the residents of the area.

Would that not by definition be a place the owner doesn't live in?

Yes. And I would suggest that we shouldn't tax renters more than owner-occupiers. Renters are already poorer on average.

Yes, and some states already charge more property tax for rentals because the owner doesn't reside primarily at the address.

If land price rises there, then it is inefficient to use the land just for a family house, instead it make sense to build there an aparment building. So land value tax works by pricing out as supposed.

Land value taxes don't make any difference there at all.

Your scenario plays out just the same. The opportunity cost difference between any two uses is exactly the same.

There's a small difference in a detail of accounting: in one case you see a higher LVT bill, in the other case you pay more for your mortgage. (Or alternatively, if you own out right, your opportunity cost for holding onto the land increases.)