I think you should just raise cost of owning additional houses so keeping them empty really hurts.
Rent control seems like government micromanagement that we want to avoid with BI.
My point is: there needs to be competition on the supply side in the housing market, to keep prices somewhat related to the real cost of the house. If this isn't possible, the government needs to step in. I hope to avoid that where possible, though it's certainly necessary to prevent landlords from raising the rent on people after they settled in a home.
Are we not talking about modern society though?
And also, you CAN build new houses in lots of places that are not the current "city centers".
Maybe higher property tax would work then? You need more revenue for UBI anyway.
Renting property definitely should be treated as a business, done by companies, and heavily controlled by regulation to favour the tenants. This is how it's done in Berlin, and it works really well.
Treating property as an investment vehicle is actively harming society.
That is banned. I don't know the limit but it's similar to what you describe. Landlords aren't allowed to raise prices by much per year. It still adds up, but by then you know to look for another place and have a year or two to do it.
If enough houses can be built, and basic income is high enough to pay free market prices, then any landlord who asks too high a price, will price themselves out of the market because there will be other cheaper houses. But for this it's vital that the supply of houses is unrestricted.
Of course a landlord might bait and switch with low rent as you move in, and rapidly increasing rent once you get settled. That's simply something that needs to be banned. After you've moved in, landlords should only be allowed to increase rent by either inflation, or the average increase in new rent in the area. But preferably just inflation.