Visualizing the Sum-Product Conjecture

Abstract

Let $SPP(n)$ be the set $\left\{\big(|A+A|,|A A|\big) : A\subseteq {\mathbb N}, |A|=n\right\}$ of sum-product pairs, where $A+A$ is the sumset $\{a+b : a,b\in A\}$ and $A A$ is the product set $\{ab:a,b\in A\}$. We construct a dataset consisting of 1162868 sets whose sum-product pairs are at least $84\%$ of $SPP(n)$ for each $n\le 32$. Notably, we do **not** see evidence in favor of Erd\H{o}s's Sum-Product Conjecture in our dataset. For $n\le 6$, we prove the exact value of $SPP(n)$. We include a number of conjectures, open problems, and observations motivated by this dataset, a large number of color visualizations.