On Lev's periodicity conjecture
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908208125181952 |
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| author | Reiher, Christian |
| author_facet | Reiher, Christian |
| contents | We classify the sum-free subsets of ${\mathbb F}_3^n$ whose density exceeds $\frac16$. This yields a resolution of Vsevolod Lev's periodicity conjecture, which asserts that if a sum-free subset ${A\subseteq {\mathbb F}_3^n}$ is maximal with respect to inclusion and aperiodic (in the sense that there is no non-zero vector $v$ satisfying $A+v=A$), then $|A|\le \frac12(3^{n-1}+1)$ -- a bound known to be optimal if $n\ne 2$, while for $n=2$ there are no such sets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_15174 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Lev's periodicity conjecture Reiher, Christian Combinatorics Number Theory 11B13, 11B30, 11P70 We classify the sum-free subsets of ${\mathbb F}_3^n$ whose density exceeds $\frac16$. This yields a resolution of Vsevolod Lev's periodicity conjecture, which asserts that if a sum-free subset ${A\subseteq {\mathbb F}_3^n}$ is maximal with respect to inclusion and aperiodic (in the sense that there is no non-zero vector $v$ satisfying $A+v=A$), then $|A|\le \frac12(3^{n-1}+1)$ -- a bound known to be optimal if $n\ne 2$, while for $n=2$ there are no such sets. |
| title | On Lev's periodicity conjecture |
| topic | Combinatorics Number Theory 11B13, 11B30, 11P70 |
| url | https://arxiv.org/abs/2408.15174 |