Visualizing the Sum-Product Conjecture

TL;DR

This paper visualizes the sum-product pairs of sets of natural numbers, constructs a large dataset covering most such pairs for small set sizes, and finds no evidence supporting Erdős's Sum-Product Conjecture within this dataset.

Contribution

The paper introduces a large dataset of sum-product pairs for small sets, visualizations, and proves exact values for small set sizes, challenging existing conjectures.

Findings

No evidence supporting Erdős's Sum-Product Conjecture in the dataset

Constructed a dataset covering at least 84% of sum-product pairs for n ≤ 32

Proved exact sum-product pairs for sets with size up to 6

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.