Visualizing the Sum-Product Conjecture

TL;DR

This work builds and analyzes a large dataset of sum-product pairs for sets of size $n\le 32$ to study Erdős's Sum-Product Conjecture. It introduces normalized visualizations of $|A+A|$ and $|AA|$ across $n$, proves exact $\mathrm{SPP}(n)$ for $n\le 6$, and presents numerous conjectures, open problems, and insights drawn from the data, including the Solymosi void and the Sidon Exclusion Zone. Despite extensive exploration, the dataset shows no persuasive evidence that the conjecture holds in practice for the tested range, and it motivates broader investigations, refined normalizations, and infrastructure for community-driven data collection. The work emphasizes the need for larger-scale experiments, domain-variant analyses, and theoretical structure (e.g., Freiman-type classifications) to resolve the conjecture's truth in a rigorous or asymptotic sense, while offering a rich resource for researchers and search engines to access concrete sum-product patterns and extremal examples.

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 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.

Figures (11)

• Figure 1: The set $\mathop{\mathrm{NSPP}}\nolimits([3,32])$. Each point represents $(|A+A|,|AA|)$ for some set $A$, normalized logarithmically to lie in $[1,2]^2$. The sets range in size from $3$ to $32$, and the color of a point reflects its cardinality, with larger sets using the red end of the rainbow. The points reflecting sets with $n=32$ elements were laid down first, then $n=31$, and so on, so that purpler points from small $n$ typically conceal redder points beneath them. The Solymosi Void is a region known to not contain any limit points (as $n\to\infty$), and SEZ is a region known not to contain any (limit or otherwise) points of $\mathop{\mathrm{NSPP}}\nolimits([3,32])$.
• Figure 2: The set $\mathop{\mathrm{NSPP}}\nolimits([3,32])$, restricted to $[1.2,1.5]\times[1.25,1.5]$. The color of the point indicates the size of the set with that normalized sum-product pair, with the reddest points corresponding to $n=32$. The $n=32$ points were laid down first, then $n=31$, and so on, so that points from small $n$ typically conceal redder points beneath them.
• Figure 3: The normalized sum-product pairs under normalizations $K,L,K^{(1)},K^{(2)}$. The reddish dots along the lower envelopes suggest that the Sum-Product Conjecture may be false, or at least the sum-product effect is not visible in sets with at most $32$ elements.
• Figure 4: The ratio of the number of pairs $(i,j)$ in $\mathop{\mathrm{SPP}}\nolimits(n)$ to the number of pairs $(i,j)$ in $[2n-1,n(n+1)/2]^2$. The bars corresponding to $n$ being a multiple of $4$ are labeled. The coloring is consistent with that used throughout this work: the rainbow covers from $n=3$ (purple) to $n=32$ (red).
• Figure 5: Plots of $\mathop{\mathrm{SPP}}\nolimits(n)$ for $n\le 6$, and a large subset of $\mathop{\mathrm{SPP}}\nolimits(n)$ for $7\le n \le 12$.

Theorems & Definitions (15)

• Conjecture 1
• Conjecture 2
• Conjecture 3
• Theorem 1: Sidon Exclusion Zone Rice
• Theorem 2: Freiman's $(3n-4)$-Theorem
• Corollary 1
• Conjecture 4
• Theorem 3
• proof
• Theorem 4