Sharp Bounds for Sets with Distinct Subset Products

Abstract

Let $A\subseteq [N]$ be such that for any pair of distinct subsets $B,C\subset A$, the products ${\prod }_{b\in B}b$ and ${\prod }_{c\in C}c$ are distinct. We prove that $|A|\le \ddot{I}€(N)+\ddot{I}€(N^{1/2})+o(\ddot{I}€(N^{1/2}))$, where $\ddot{I}€$ is the prime counting function, answering a question of Erdős.

13 pages, 4 figures. Updated to match published version