Affine Equivalence of Subsets of $\mathbb{F}_2^n$ via Venn Diagrams and Applications to Sidon Sets
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911131757445120 |
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| author | Calta, Kariane Covey, Sarah Goldberg, Timothy E. Rose, Lauren L. Rose-Levine, Daniel |
| author_facet | Calta, Kariane Covey, Sarah Goldberg, Timothy E. Rose, Lauren L. Rose-Levine, Daniel |
| contents | Two subsets $S$ and $T$ of $\mathbb{F}_2^n$ are \textit{affinely equivalent} if there is an affine automorphism of $\mathbb{F}_2^n$ taking $S$ to $T$. Given a basis of the affine span of $S$, we can construct a Venn diagram whose regions partition $S$. We prove that any two bases of $\operatorname{aff}(S)$ will have the same Venn diagram up to a linear permutation of the Venn regions. Moreover, we prove that two sets are affinely equivalent if and only if there is a cardinality-preserving linear permutation from the Venn regions of $S$ to the Venn regions of $T$. We use these results to classify certain Sidon sets up to affine equivalence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00556 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Affine Equivalence of Subsets of $\mathbb{F}_2^n$ via Venn Diagrams and Applications to Sidon Sets Calta, Kariane Covey, Sarah Goldberg, Timothy E. Rose, Lauren L. Rose-Levine, Daniel Combinatorics 05B25 (Primary) 51E10, 05B05 (Secondary) Two subsets $S$ and $T$ of $\mathbb{F}_2^n$ are \textit{affinely equivalent} if there is an affine automorphism of $\mathbb{F}_2^n$ taking $S$ to $T$. Given a basis of the affine span of $S$, we can construct a Venn diagram whose regions partition $S$. We prove that any two bases of $\operatorname{aff}(S)$ will have the same Venn diagram up to a linear permutation of the Venn regions. Moreover, we prove that two sets are affinely equivalent if and only if there is a cardinality-preserving linear permutation from the Venn regions of $S$ to the Venn regions of $T$. We use these results to classify certain Sidon sets up to affine equivalence. |
| title | Affine Equivalence of Subsets of $\mathbb{F}_2^n$ via Venn Diagrams and Applications to Sidon Sets |
| topic | Combinatorics 05B25 (Primary) 51E10, 05B05 (Secondary) |
| url | https://arxiv.org/abs/2509.00556 |