one of those is not like the other. Berlin has recently passed a law capping rent at below 9 Euro per square meter (much less in many regions) all over the city.
It's the first time such a law was passed anywhere in Germany. [1]
[1] https://www.spiegel.de/wirtschaft/soziales/eckpunkte-fuer-mi...
What if it's the reverse?
The supply of land is fixed. Its intensity of usage is not.
By imposing a per unit rent cap, what's effectively happening is that a rule is being established that no more rent than that cap can be extracted per unit. But there's no inherent restriction (zoning, construction laws, etc., being of course the exceptions) to more intensive development. Increasing the unit count.
A rent cap plus zoning restrictions is one form of death: there's no new housing supplied, and if the issue is rent control (limited rates of increase over tenancy), then the result is similar to what happens in San Francisco.
But with sane zoning and construction standards and a cap, you're inducing more intensive development -- an increase in housing supply -- by imposing an arbitrary per-unit revenue cap. And without nominally raising taxes, the bugbear of the land value tax approach.
I'm not aware this argument has ever been proposed, though I am fairly confident it has been.
Your hypothesis assumes that increasing the potential revenue by square meter has no effect attracting Urban development projects.
It seems obvious that if it's possible to increase intensity then being able to generate more income per m² only helps put together a valid business plan that increases supply.
There are two ways to increase revenue per m^2. One is to intensify the construction. The other is to limit supply.
For numerous reasons, the latter approach ... seems to predominate. Rather than constructing denser residential, commercial, and (where appropriate) industrial space, the tendency is both to sprawl out, and to simply refuse to allow new construction. San Francisco is the pathalogical case, but hardly the only, and numerous other US growth areas (LA, Seattle, Austin) and elsewhere (Sydney, Vancouver, London) are to large degrees similar. There's new construction ... of large McMansions. But not of denser forms of housing.
By imposing an arbitrary constraint in one dimension (rent extraction per unit) but allowing intensification on another (density or height of construction), this should be addressable.
Keep in mind that most of SF is at best 2 storey construction. You don't have to airdrop Salesforce Towers all over the city, only allow construction to 4-5 storeys, to more than double effective density. Apply this more broadly over the SF Bay Area, and what's become a region-threatening blockage to housing availability (and incidental issues such as traffic congestion, hours-long commutes, access to schools and services, etc., etc.) would be lifted.
The fact that landlords (or banks, through mortgages) can extract arbitrarily high amounts of rent from a given area regardless of intensity of development, is the problem. The land value tax is one approach, and is probably a better one. I'm considering what the implications of a rents cap might be.
A usual objection to rent controls (increases on existing units) is that landlords are not incentivised to maintain or improve the quality of units. Whether that might also apply to a cap is a possible objection.