Elementary solution to the fair division problem

The paper proves an envy-free (strong) division for atomless charges via an explicit iterative scheme that carves out, at each macro-step, a sizeable subset H on which a strong solution exists, and shows exponential decay of the remainder with respect to the summed measure µ = Σi µi, namely µ(Ms) ≤ ((2^{r−1}−1)/2^{r−1})^s µ(M) for the measures case, and an analogous decay in the charges case, stitching positive/negative parts through Hahn–Jordan and a gentleman’s division construction. These claims are established in Theorem 2.1 and the constructions of Sections 5–6, with the decay bound explicitly derived in Corollary 5.3 and Lemma 6.5/Corollary 6.6 . The model solution gives a different, Lyapunov-based “equal-clone” algorithm: repeatedly halve vector measures to create 2^{r−1} equal clones and assign r of them each round; this yields a stronger property (exact division from every agent’s perspective, not just envy-freeness) and a sharper rate (for measures, 1−r/2^{r−1} per round). Both approaches rely on atomlessness and Lyapunov convexity (explicitly assumed or invoked) and are coherent. Thus, both are correct, with substantially different constructions and quantitative guarantees.