The sum-product problem for small sets II

TL;DR

This paper proves new lower bounds on the number of distinct pairwise sums or products for small sets of natural numbers, extending previous results and classifying extremal sets with specific sum/product properties.

Contribution

It establishes exact thresholds for the number of sums or products in sets of size 10 and 11, and classifies extremal sets, extending prior work for smaller sets.

Findings

Sets of size 10 determine at least 30 sums or products.

Sets of size 11 determine at least 34 sums or products.

Unique extremal sets are identified for these thresholds.

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.