The sum-product problem for small sets II

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 $k\le 9$. 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.

14 pages, 5 figures, 1 table, 3 algorithms, uniqueness established in Section 5 with Theorem 1.5 updated accordingly, Lemma 3.1 statement corrected, typos corrected