In this post you said, that with the shares / basic income you get you can take part in auctions to buy land. But this would be only possible if the land isnt yet distributed. Therefore after the first distribution of land there wouldn't be any auction at all anymore, therefore the basic income would most likely be worth less.

How can we make it fair for new participants?
To solve that lets now imagine we would continue with the auctions. If someone in the future is ready to pay more, the land goes to him, and the old owner gets the money. This solution would be fair for the new participants, but it doesn't sound fair for the investors, that invest their time and money to improve the land.

How could we make it fair for investors?
We could require the new owner to pay for example double the amount of money then the old owner. In case of low investments this would help, but if someone invests a lot more in the land he could still loose a lot.

Using stock bases companies for fair property management
To solve that we could allow the owner to put his land in a stock based cooperation that is secured from auctions. To be fair for future participants the new stocks must increase with the same inflation as the basic income increases per year. These new stocks must then be sold on an open transparent market in exchange for the basic income currency. To be fair the earnings from this must then be either destroyed, or given out to all participants equally on top of the basic income.

The same which is true for land should also be true for all other property rights that one can claim like land, patents, buildings....

To simplify it each participant could have a certain amount of land that he cannot loose through auctions. Participants could group up to secure bigger amount of lands.

But what is now if someone buys a land for a far too high price from himself, just to protect his land from others? In this case the price would not really reflect the true price and therefore hinder others to take part in the auctions for the land. To solve that, we could require, that an certain amount of the profit of the auction, lets say 50% is simply destroyed. So one could pay more, to still claim his land, but 50% would be to the benefit of all.
This would also greatly reflect that some part of the profit from using the resources belongs to all.

Summary

• To be fair for new participants auctions of property should go on in the future
• To be fair for investors the new owner should pay more, lets say double then the old one
• To hinder abuse and to reflect that in the end the resources belong to all, some part of the profit from the auction should be given to all participants
• To be fair a certain amount of land / property per participant could be free from further auctions
• To secure big investments, land / property rights could be put in an stock based company
• to be fair for all participants new stocks equaly to the increase of the basic income should be given out per year and sold on an open market
• To be fair 50% of the profit for selling these stocks should be given to all participants