Tight Lower Bounds for Unequal Division

Alice and Bob want to cut a cake and have their own opinions of what values various pieces have. In contrast to the usual problems of fair division, they want to cut it unfairly. More precisely, they want to cut it in ratio (a, b). (We can assume gcd(a,b)=1.) Let f(a, b) be the number of cuts this will take (assuming both act in their own self interest). It is known that f(a, b) ≤ dlg(a + b)e [1]. We show that 1. for all a,b, f(a, b) ≥ lg(lg(a + b)) 2. for an infinite number of (a,b), f(a, b) ≤ 1 + lg(lg(a + b)).