Sharp Bounds for Sets with Distinct Subset Products

TL;DR

This paper establishes an upper bound on the size of sets with unique subset products, using prime counting functions, thereby answering a question posed by Erdős.

Contribution

It provides the first sharp bounds for sets with distinct subset products, linking combinatorial properties to prime number theory.

Findings

Bound on set size: |A| ≤ π(N)+π(N^{1/2})+o(π(N^{1/2}))

Answer to Erdős's question on subset product uniqueness

Connection between subset products and prime counting functions

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 $|A|\leq \pi(N)+\pi(N^{1/2})+o(\pi(N^{1/2}))$, where $\pi$ is the prime counting function, answering a question of Erd\H{o}s.