Visualizing the Sum-Product Conjecture

Abstract

Let $SPP(n)$ be the set $\left\{(|A+A|,|AA|):A\subseteq \mathbb{N},|A|=n\right\}$ 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 $84\%$ 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.

28 pages, many color images