Sorry if what you explain makes me not understand
Not a problem.
1. Each address has a right to claim 1 token. This means that there are (in theory only) infinite number of claims that anyone can make.
2. Each claim costs some GAS money (this means that I am thinking about Ethereum network or any other where you have to pay the GAS).
3. We can automatically regulate the amount of GAS money increase per time, number of tokens claimed, or both. In a sense this would be equivalent to a constantly-rising mining difficulty... whether you spend the GAS money or pay the electricity bill... does not matter at the end. We could complicate it more and make it follow the actual mining difficulty of the main network currency, but lets keep it simple for now.