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arXiv · 2509.00556

Affine Equivalence of Subsets of $\mathbb{F}_2^n$ via Venn Diagrams and Applications to Sidon Sets

Kariane Calta·Sarah Covey·Timothy E. Goldberg·Lauren L. Rose·Daniel Rose-Levine
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Abstract

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.

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Kariane Calta, Sarah Covey, Timothy E. Goldberg, Lauren L. Rose, Daniel Rose-Levine. 2025-08-30. Affine Equivalence of Subsets of $\mathbb{F}_2^n$ via Venn Diagrams and Applications to Sidon Sets. https://arxiv.org/abs/2509.00556

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