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

## TL;DR

This paper introduces a method to determine affine equivalence of subsets in $\

## Contribution

It establishes a Venn diagram-based characterization of affine equivalence and applies it to classify Sidon sets up to affine transformations.

## Key findings

- Venn diagrams of affine span bases are equivalent up to linear permutation.
- Affine equivalence corresponds to cardinality-preserving linear permutations of Venn regions.
- Classification of certain Sidon sets up to affine equivalence.

## 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.