Integer sets with distinct subset-sums

Abstract:

In Section 1 we introduce the problem of finding minimal-height sets of n natural numbers with distinct subset-sums (SSD), and in Section 2 review the well-known Conway-Guy sequence u, conjectured to yield a minimal SSD set for every n. We go on (Section 3) to prove that u certainly cannot be improved upon by any "greedy" sequence, to verify numerically (Section 4) that it does yield SSD sets for $n < 80$, and (Section 5) by direct search to show that these are minimal for $n \leqslant 8$. There is a brief interlude (Section 6) on the problem of decoding the subset from its sum. In Section 7 generalizations of u are constructed which are asymptotically smaller: Defining the Limit Ratio of a sequence w to be $\alpha = {\lim _{n \to \infty }}{w_n}/{2^{n - 1}}$, the Atkinson-Negro-Santoro sequence v (known to give SSD sets) has $\alpha = 0.6334$, Conway-Guy (conjectured to) has $\alpha = 0.4703$, and our best generalization has $\alpha = 0.4419$. We also (Section 8) discuss when such sequences have the same $\alpha$, and (Section 9) how $\alpha$ may efficiently be computed to high accuracy.