Could a tax on rental income that depends on price per square meter induce more housing with lower rents?
Why do you think that additional taxes on more expensive apartments (progressive taxes) would lead to lowering the prices and offering them cheaper than market dictates? 

Adding taxes on top would just make these apartments even more expensive for the consumer, since the owner will push down the additional cost to the consumer rather than take a haircut. Especially if these apartments are laying around empty anyway. 

If I am willing to sell product A for say 1.000$ but there are no buyers - will an increase in the tax help me sell it? 
If you on the other hand gave me a tax discount, I might want to sell it cheaper, since I may be left with the same after-tax profit.