Visualizing the Sum-Product Conjecture

Let $SPP(n)$ be the set $\left\{\right(\mathrm{\mid }A+A\mathrm{\mid },\mathrm{\mid }AA\mathrm{\mid }):A\subseteq \mathbb{N},\mathrm{\mid }A\mathrm{\mid }=n\}$ of sum-product pairs, where $A+A$ is the sumset $\{a+b:a,b\in A\}$ and $AA$ is the product set $\{ab:a,b\in A\}$. We construct a dataset consisting of 1162868 sets whose sum-product pairs are at least $84\mathrm{\%}$ of $SPP(n)$ for each $n\le 32$. Notably, we do not see evidence in favor of Erdős’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.