The sum-product problem for small sets II

TL;DR

This work resolves the exact sum-product thresholds for small sets by proving $SP(10)=30$ and $SP(11)=34$, with the extremal configurations unique up to scaling. It combines a log-transform approach to convert products into sums, Freiman-type structure theory, and intensive computational classification to show that near-extremal sets must lie in one of two two-dimensional geometric progression families (up to scaling), or in a single geometric progression. A detailed collision analysis within these GP frameworks yields sharp lower bounds on $|A+A|$, ruling out all but the known extremal examples for $|A|=10$ and $|A|=11$. The paper also provides a comprehensive classification of small-product configurations, introduces algorithmic tools (WinnersSearch) and resultants-based pruning for collision detection, and outlines future work toward higher $k$, including a conjectured $SP(12)=41$ and a higher-dimensional Freiman-type theory.

Abstract

We establish that every set of $k=10$ natural numbers determines at least $30$ distinct pairwise sums or at least $30$ distinct pairwise products, as well as the analogous result for $k=11$ and at least $34$ sums/products, with sharpness uniquely (up to scaling) exhibited by $\{1, 2, 3, 4, 6, 8, 9, 12, 16, 18\}$ and $\{1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24\}$, respectively. This extends previous work of the fifth author with Clevenger, Havard, Heard, Lott, and Wilson, which established the corresponding thresholds for $k\leq 9$. Included is a classification result for sets of $10$ real numbers (resp. positive real numbers) determining at most $29$ pairwise sums (resp. pairwise products) that do not contain $8$ elements of any single arithmetic progression (resp. geometric progression), as well as some observations controlling additive quadruples in small subsets of two-dimensional generalized geometric progressions.

Figures (5)

• Figure 1: One example from items (i)-(iii), and unique structure from (iv), in Lemma \ref{['1029sum']}.
• Figure 2: Example from item (ii), and unique structure from (iii), in Corollary \ref{['1029sumcor']}.
• Figure 3: The red and blue boxes indicate the configurations from items (2) and (3), respectively, in Corollary \ref{['sumcornew']}.
• Figure 4: Single and double two-row collisions.
• Figure 5: Single and double three-row collisions, showing maximum of $6$ collisions in $G_2$.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark
• Lemma 1.4
• Theorem 1.5
• Lemma 3.1
• Lemma 3.2
• Remark
• Corollary 3.3