3/4 bdrm (1600-2000 sf) rent-controlled apts (flats) in SF typically rent for $2000 – $2500 mo. That is the cheapest I’ve heard of and SF’s rents are probably on par with Manhattan (NY, NY’s) rents.
There are two levels of rent control in NYC (not just Manhattan). “Rent stabilized” units fetch the kind of rents that you mention, up to a max rent of $2500.
“Rent controlled” units are much, much cheaper, but they’re also rare, since they stopped “making” new ones in the 70s, and only units that have remained continuously in a family are allowed to continue being controlled.