The sum-product problem for small sets II
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914371337191424 |
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| author | Antis, Phillip Britt, Holden Chapman, Caleigh Hawkins, Elizabeth Rice, Alex Warren, Elyse |
| author_facet | Antis, Phillip Britt, Holden Chapman, Caleigh Hawkins, Elizabeth Rice, Alex Warren, Elyse |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21828 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The sum-product problem for small sets II Antis, Phillip Britt, Holden Chapman, Caleigh Hawkins, Elizabeth Rice, Alex Warren, Elyse Combinatorics Number Theory 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. |
| title | The sum-product problem for small sets II |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2601.21828 |