# Lipschitz bijections between boolean functions

## TL;DR

This paper investigates Lipschitz bijections between boolean functions, providing new bounds, constructions, and embeddings that clarify the limitations and possibilities of such mappings.

## Contribution

It offers new results on the existence and properties of Lipschitz bijections between key boolean functions, including bounds and explicit constructions.

## Key findings

- No $O(1)$-bi-Lipschitz bijection from Dictator to XOR with $O(1)$ output dependence.
- Constructed XOR to Majority mapping with average stretch $O(
oot n)$.
- Embedded XOR into Majority with a 3-Lipschitz map in $2n+1$ dimensions.

## Abstract

We answer four questions from a recent paper of Rao and Shinkar on Lipschitz bijections between functions from $\{0,1\}^n$ to $\{0,1\}$. (1) We show that there is no $O(1)$-bi-Lipschitz bijection from $\mathrm{Dictator}$ to $\mathrm{XOR}$ such that each output bit depends on $O(1)$ input bits. (2) We give a construction for a mapping from $\mathrm{XOR}$ to $\mathrm{Majority}$ which has average stretch $O(\sqrt{n})$, matching a previously known lower bound. (3) We give a 3-Lipschitz embedding $\phi : \{0,1\}^n \to \{0,1\}^{2n+1}$ such that $\mathrm{XOR}(x) = \mathrm{Majority}(\phi(x))$ for all $x \in \{0,1\}^n$. (4) We show that with high probability there is a $O(1)$-bi-Lipschitz mapping from $\mathrm{Dictator}$ to a uniformly random balanced function.