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.