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

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 $\mathrm{aff}(S)$ 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.

33 pages, 5 figures, 8 tables