250 problems in elementary number theory by Waclaw Sierpinski

By Waclaw Sierpinski

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It is obvious that the function m from Part to the IPS respects s-equivalence. (That is, if partitions P and Q are s-equivalent, then m(P) = m(Q)). We are interested in whether or not m is one-to-one for non-s-equivalent partitions. Or, equivalently, we are interested in whether the function induced by m that maps s-classes of Part to the IPS is one-to-one. 6 Let p be a point of the IPS. The following are equivalent: a. p is the image, under m, of infinitely many mutually non-s-equivalent partitions.

The Situation Without Absolute Continuity 43 c. There are infinitely many strongly c-proportional s-classes. d. There are infinitely many strongly c-proportional p-classes. Next, we reconsider our results on efficiency from Section 3B. 8 holds regardless of any assumptions about absolute continuity. 9, which told us that the outer Pareto boundary is equal to the outer boundary and the inner Pareto boundary is equal to the inner boundary. 9. 10, our present assumption, together with the fact that the IPS is symmetric about ( 12 , 12 ), implies that the IPS includes points of the form (1, a) and (0, 1 − a), or else points of the form (a, 1) and (1 − a, 0), for some a with 0 < a < 1.

Hence, p lies in the interior of a line segment contained in the IPS. Next, we show that part c implies part a. Suppose that p is a point that lies in the interior of some line segment contained in the IPS. Let P = P1 , P2 and Q = Q 1 , Q 2 be two partitions such that p is the midpoint of the line segment connecting m(P) and m(Q). We can imagine Q as being obtained from P by two transfers of cake, one from Player 1 to Player 2, and one from Player 2 to Player 1. Since p is the midpoint of the line segment connecting m(P) and m(Q), p = m(R) where R is any partition obtained by completing “half” of each of these two transfers.

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