Belt Auction - What law stated that if you settled on some land and improved it, you could purchase that land at a minimum?

What law stated that if you settled on some land and improved it, you could purchase that land at a minimum?

At a minimum price without a competitive bid?

You are talking about Homestead laws, which have been used in many places to encourage settlement. They only applied to specific government owned land and have been repealed long ago..
There have been homestead laws or you could be asking about adverse possession.
That would be a "homestead act", i.e., a government handout to those willing to settle and work the land. It has NOTHING to do with "adverse possession", which relates to trespassing and taking legal title without having paid a dime or done any work (other than perhaps keeping others off the land you claim as yours, although a few jurisdictions require orderly payment of property taxes in order to claim title).