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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.01665 |
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| _version_ | 1866917974423633920 |
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| author | Bettin, Sandro Bordignon, Matteo Fazzari, Alessandro |
| author_facet | Bettin, Sandro Bordignon, Matteo Fazzari, Alessandro |
| contents | In a previous work, Bettin, Koukoulopoulos, and Sanna prove that if two sets of natural numbers $A$ and $B$ have natural density $1$, then their product set $A \cdot B := \{ab : a \in A, b \in B\}$ also has natural density $1$. They also provide an effective rate and pose the question of determining the optimal rate. We make progress on this question by constructing a set $A$ of density 1 such that $A\cdot A$ has a ''large'' complement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01665 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On products of sets of natural density one Bettin, Sandro Bordignon, Matteo Fazzari, Alessandro Number Theory 11B05 In a previous work, Bettin, Koukoulopoulos, and Sanna prove that if two sets of natural numbers $A$ and $B$ have natural density $1$, then their product set $A \cdot B := \{ab : a \in A, b \in B\}$ also has natural density $1$. They also provide an effective rate and pose the question of determining the optimal rate. We make progress on this question by constructing a set $A$ of density 1 such that $A\cdot A$ has a ''large'' complement. |
| title | On products of sets of natural density one |
| topic | Number Theory 11B05 |
| url | https://arxiv.org/abs/2504.01665 |