Fair division with unequal shares is an intensively studied resource allocation problem. For i∈[n]i∈[n], let μiμi be an atomless probability measure on the measurable space (C,S)(C,S) and let titi be positive numbers (entitlements) with ∑ni=1ti=1∑i=1nti=1. A fair division is a partition of CC into sets Si∈SSi∈S with μi(Si)≥tiμi(Si)≥ti for every i∈[n]i∈[n].
We introduce new algorithms to solve the fair division problem with irrational entitlements. They are based on the classical Last diminisher technique and we believe that they are simpler than the known methods. Then we show that a fair division always exists even for infinitely many players.