An inverse theorem for sumsets of sets of positive density in the integers
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908964063543296 |
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| author | Ackelsberg, Ethan Richter, Florian K. |
| author_facet | Ackelsberg, Ethan Richter, Florian K. |
| contents | Let $d(\cdot)$ denote the natural density on the positive integers. We characterize all sets $A,B$ with positive density satisfying $d(A+B)=d(A)+d(B)$, under the assumption that the two sets are not both contained in a proper finite union of residue classes. This gives a new inverse theorem for Kneser's sumset inequality in the integers, and provides a partial answer to a long-standing open question of Erdős and Graham. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12864 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An inverse theorem for sumsets of sets of positive density in the integers Ackelsberg, Ethan Richter, Florian K. Number Theory Combinatorics Dynamical Systems 11P70 (Primary) 11B30, 37A44 (Secondary) Let $d(\cdot)$ denote the natural density on the positive integers. We characterize all sets $A,B$ with positive density satisfying $d(A+B)=d(A)+d(B)$, under the assumption that the two sets are not both contained in a proper finite union of residue classes. This gives a new inverse theorem for Kneser's sumset inequality in the integers, and provides a partial answer to a long-standing open question of Erdős and Graham. |
| title | An inverse theorem for sumsets of sets of positive density in the integers |
| topic | Number Theory Combinatorics Dynamical Systems 11P70 (Primary) 11B30, 37A44 (Secondary) |
| url | https://arxiv.org/abs/2604.12864 |